Abstract: The aim of this paper is to exhibit the research on separation axioms in terms of nearly open sets viz
p-open, s-open, α-open & β-open sets. It contains the topological property carried by respective ℘ -Tk spaces (℘
= p, s, α & β; k = 0,1,2) under the suitable nearly open mappings . This paper also projects ℘ -R0 & ℘ -R1
spaces where ℘ = p, s, α & β and related properties at a glance. In general, the ℘ -symmetry of a topological
space for ℘ = p, s, α & β has been included with interesting examples & results.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
The determination of this paper is to introduce two new spaces , namely 𝑆𝑔
∗
-compact and 𝑆𝑔
∗
-
connected spaces. Additionally some properties of these spaces are investigated.
Mathematics Subject Classification: 54A05
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
The determination of this paper is to introduce two new spaces , namely 𝑆𝑔
∗
-compact and 𝑆𝑔
∗
-
connected spaces. Additionally some properties of these spaces are investigated.
Mathematics Subject Classification: 54A05
Using ψ-Operator to Formulate a New Definition of Local Functionpaperpublications3
Abstract:In this paper, we use ψ-operator in order to get a new version of local function. The concepts of maps, dense, resolvable and Housdorff have been investigated in this paper, as well as modified to be useful in general الخلاصة: استخدمنافيهذاالبحثالعمليةابسايمنأجلالحصولعلىنسخهجديدهمنالدالةالمحلية.وقدتمالتحقيقفيمفاهيمالدوال, الكثافة,المجموعاتالقابلةللحلوفضاءالهاوسدورف,وكذلكتعديلهالتكونمفيدةبشكلعام. Keywords:ψ-operator, dense, resolvable and Housdorff.
Title:Using ψ-Operator to Formulate a New Definition of Local Function
Author:Dr.Luay A.A.AL-Swidi, Dr.Asaad M.A.AL-Hosainy, Reyam Q.M.AL-Feehan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
In the present paper , we introduce and study the concept of gr- Ti- space (for i =0,1,2) and
obtain the characterization of gr –regular space , gr- normal space by using the notion of gr-open
sets. Further, some of their properties and results are discussed.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
If X be a topological space and A subspace of X, then a point x E X is said to be a cluster point of A if every open ball centered at x contains at least one point of A different from X. In the preliminary sections, review of the interior of the set X was discussed before the major work of section three was implemented.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Using ψ-Operator to Formulate a New Definition of Local Functionpaperpublications3
Abstract:In this paper, we use ψ-operator in order to get a new version of local function. The concepts of maps, dense, resolvable and Housdorff have been investigated in this paper, as well as modified to be useful in general الخلاصة: استخدمنافيهذاالبحثالعمليةابسايمنأجلالحصولعلىنسخهجديدهمنالدالةالمحلية.وقدتمالتحقيقفيمفاهيمالدوال, الكثافة,المجموعاتالقابلةللحلوفضاءالهاوسدورف,وكذلكتعديلهالتكونمفيدةبشكلعام. Keywords:ψ-operator, dense, resolvable and Housdorff.
Title:Using ψ-Operator to Formulate a New Definition of Local Function
Author:Dr.Luay A.A.AL-Swidi, Dr.Asaad M.A.AL-Hosainy, Reyam Q.M.AL-Feehan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
In the present paper , we introduce and study the concept of gr- Ti- space (for i =0,1,2) and
obtain the characterization of gr –regular space , gr- normal space by using the notion of gr-open
sets. Further, some of their properties and results are discussed.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
If X be a topological space and A subspace of X, then a point x E X is said to be a cluster point of A if every open ball centered at x contains at least one point of A different from X. In the preliminary sections, review of the interior of the set X was discussed before the major work of section three was implemented.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Survey on Threats and Security schemes in Wireless Sensor NetworksIJERA Editor
It is difficult to achieve and become particularly acute in wireless sensor networks due to the limitation in network capability, computational power and memory which do not allow for implementation of complex security mechanism because security being vital to the acceptance and use of wireless sensor networks for many applications. In this paper we have explored general security threats in wireless sensor networks and analyzed them. This paper is an attempt to survey and analyze the threats to the wireless sensor networks and focus on the type of attacks and achieve secure communication in wireless sensor networks.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Memory Polynomial Based Adaptive Digital PredistorterIJERA Editor
Digital predistortion (DPD) is a baseband signal processing technique that corrects for impairments in RF
power amplifiers (PAs). These impairments cause out-of-band emissions or spectral regrowth and in-band
distortion, which correlate with an increased bit error rate (BER). Wideband signals with a high peak-to-average
ratio, are more susceptible to these unwanted effects. So to reduce these impairments, this paper proposes the
modeling of the digital predistortion for the power amplifier using GSA algorithm.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Literature Review on Lean Implementations – A comprehensive summaryIJERA Editor
The available research papers in area of Lean are studied to know the implementation level of different lean
tools, barrier and benefits of implementation are also considered in the review .The commonly used lean tools in
the various organization, most common barriers and benefits have been identified and listed in this paper. Most
common barrier are also components of quality of work life.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
All people want to improve their quality of life and this can be achieved only by technology. Many problems are faced by daily vehicle users in payment based parking systems, both in open air parking system where the parking is done along the streets and in closed parking system where parking is done in closed infrastructure added with entry and exit points. Delays (long queues) and accuracy in fares are the main problems faced by the users. Many solutions are proposed to solve this problem but all have their own drawbacks. In this paper a new solution is proposed based on Near Field Communication (NFC) which makes the payment system reliable and easy.
Implementation of Transformer Protection by Intelligent Electronic Device for...IJERA Editor
Protection of power system equipments was traditionally done by using electromagnetic relay, static relays, and
numerical relays. At present the microprocessor based relays are replacing the old Electromagnetic relays
because of their high level accuracy and fast operation. RET670(Transformer protection relay ), an IED
(INTELLIGENT ELECTRONIC DEVICE) provides fast and selective protection, monitoring, and control of all
types of transformer. The configured IED is tested under different fault conditions simulated by using mobile test
kit to ensure IED’s reliable operation on site. With preconfigured algorithms, the IED will automatically
reconfigure the network in case of a fault, and a service restoration is carried out within milliseconds by giving
trip signal to the corresponding Circuit breakers. On receiving the trip signal the circuit breaker operates
providing quicker isolation of transformers under the fault condition. This enables to have a complete and an
adequate protection to the specified power transformer.
Grapheme-To-Phoneme Tools for the Marathi Speech SynthesisIJERA Editor
We describe in detail a Grapheme-to-Phoneme (G2P) converter required for the development of a good quality
Marathi Text-to-Speech (TTS) system. The Festival and Festvox framework is chosen for developing the
Marathi TTS system. Since Festival does not provide complete language processing support specie to various
languages, it needs to be augmented to facilitate the development of TTS systems in certain new languages.
Because of this, a generic G2P converter has been developed. In the customized Marathi G2P converter, we
have handled schwa deletion and compound word extraction. In the experiments carried out to test the Marathi
G2P on a text segment of 2485 words, 91.47% word phonetisation accuracy is obtained. This Marathi G2P has
been used for phonetising large text corpora which in turn is used in designing an inventory of phonetically rich
sentences. The sentences ensured a good coverage of the phonetically valid di-phones using only 1.3% of the
complete text corpora.
Implementation and Performance Analysis of Kaiser and Hamming Window Techniqu...IJERA Editor
Considering the importance of real time filtering, a comparison was done between two prominent window
techniques known for digital filtering. A 16 tap digital band pass FIR filter is designed for each design technique
(Kaiser and Hamming) and implemented over FPGA. The Simulink model of the filter confirms the correctness
and other properties of the digital filter. Further a hardware descriptive code (VHDL) is generated for the
designed filter which then will be loaded on to the FPGA. The VHDL code is speed optimized. The VHDL code
is simulated and synthesized in Xilinx ISE. Further the performance analysis is done on FPGA to determine the
applicability of the filter.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
μ-πrα Closed Sets in Bigeneralized Topological SpacesIJERA Editor
The aim of the paper is to introduce the concept of μ(m,n)-πrα closed sets in bigeneralized topological spaces and study some of their properties. We also introduce the notion of μ(m,n)-πrα continuous function and μ(m,n)-πrα T1/2 spaces on bigeneralized topological spaces and investigate some of their properties. Mathematics subject classification: 54A05, 54A10
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the
conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
Continuous functions play a dominant role in analysis and homotopy theory. They
have applications to image processing, signal processing, information, statistics,
engineering and technology. Recently topologists studied the continuous like functions
between two different topological structures. For example, semi continuity between a
topological structure, α-continuity between a topology and an α-topology.
Nithyanantha Jothi and Thangavelu introduced the concept of binary topology in
2011. Recently the authors extended the notion of binary topology to n-ary topology
where n˃1 an integer. In this paper continuous like functions are defined between a
topological and an n-ary topological structures and their basic properties are
studied.
The maiin objjectt off tthiis paper tto iinttroduce T--pre--operattor,,T--pre--open sett,, T--pre--monottone,,pre--
subadiittiive operattor and pre--regullar operattor.. As wellll as we iinttroduce ((T,,L))pre--conttiinuiitty.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
In this paper, we define and study about a new type of generalized closed set called, 𝑔∗s-closed set.Its relationship with already defined generalized closed sets are also studied.
Similar to An Overview of Separation Axioms by Nearly Open Sets in Topology. (20)
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Final project report on grocery store management system..pdf
An Overview of Separation Axioms by Nearly Open Sets in Topology.
1. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 97 | P a g e
An Overview of Separation Axioms by Nearly Open Sets in
Topology.
Dr. Thakur C. K. Raman, Vidyottama Kumari
Associate Professor & Head, Deptt. Of Mathematics, Jamshedpur Workers,
College, Jamshedpur, Jharkhand.
INDIA.
Assistant Professor, Deptt. Of Mathematics, R.V.S College OF Engineering & Technology, Jamshedpur,
Jharkhand, INDIA.
Abstract: The aim of this paper is to exhibit the research on separation axioms in terms of nearly open sets viz
p-open, s-open, α-open & β-open sets. It contains the topological property carried by respective ℘ -Tk spaces (℘
= p, s, α & β; k = 0,1,2) under the suitable nearly open mappings . This paper also projects ℘ -R0 & ℘ -R1
spaces where ℘ = p, s, α & β and related properties at a glance. In general, the ℘ -symmetry of a topological
space for ℘ = p, s, α & β has been included with interesting examples & results.
Key Words : ℘ -Tk spaces, ℘ -R0 & ℘ -R1 spaces & ℘ -symmetry .
I. Introduction & Preliminaries:
The weak forms of open sets in a topological space as semi-pre open & b-open sets were introduced by D.
Andrijevic through the mathematical papers[1,2]. The concepts of generalized closed sets with the introduction
of semi-pre opens were studied by Levine [12] and Njasted [14] investigated α-open sets and Mashour et. al.
[13] introduced pre-open sets. The class of such sets is named as the class of nearly open sets by Njasted[14].
After the works of Levine on semi-open sets, several mathematician turned their attention to the
generalization of various concepts of topology by considering semi-open sets instead of open sets. When open
sets are replaced by semi-open sets, new results were obtained. Consequently, many separation axioms have
been formed and studied.
The study of topological invariants is the prime objective of the topology. Keeping this in mind several
authors invented new separation axioms. The presented paper is the overview of the common facts of this trend
at a glance for researchers.
Throughout this paper, spaces (X, T) and (Y,σ) (or simply X and Y) always mean topological spaces on
which no separation axioms are assumed unless explicitly stated. The notions mentioned in
[1,6],[2],[12],[14]&[13] were conceptualized using the closure operator (cl) & the interior operator( int ) in the
following manner:
Definition:
A subset A of a topological space (X,T) is called
I. a semi-pre –open[1] or β-open [6] set if A⊆ cl(int(cl(A))) and a semi-pre closed or β-closed if
int(cl(int(A))) ⊆ A.
II. a b-open[2] set if A⊆ cl(int(A))∪ int(cl(A)) and a b-closed [8] if cl(int(A)) ∩int(cl(A)) ⊆A.
III. a semi-open [12] set if A⊆ cl(int(A)) and semi-closed if int(cl(A)) ⊆ A.
IV. an α-open[14] set if A⊆ int(cl(int(A))) and an α-closed set if cl(int(cl(A))) ⊆ A.
V. a pre-open [13] set if A ⊆ int(cl (A)) and pre-closed if cl(int(A)) ⊆ A.
The class of pre-open, semi-open ,α –open ,semi-pre open and b-open subsets of a space (X,T) are usually
denoted by PO(X,T),SO(X,T),
T , SPO(X,T) & BO(X,T) respectively. Any undefined terminology used in
this paper can be known from [4].
In 1996, D.Andrijevic made the fundamental observation:
Proposition:
For every space (X,T) , PO(X,T) ∪ SO(X,T) ⊆ BO(X,T) ⊆ SPO(X,T) holds but none of these
implications can be reversed[10].
RESEARCH ARTICLE OPEN ACCESS
2. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 98 | P a g e
Proposition : Characterization [10]:
I. S is semi-pre-open iff S ⊆ sint(sclS).
II. S is semi-open iff S ⊆ scl (sintS) .
III. S is pre-open iff S ⊆ pint(pclS) .
IV. S is b-open iff S ⊆ pcl(pint S). , where S⊆ X & (X,T) is a space.
The separation axioms of topological spaces are usually denoted with the capital letter “T” after the
German word “Trennung” which means separation. Separation axioms are one among the most common
important & interesting concepts in Topology and are used to coin more restricted classes of topological spaces.
However, the structure and the properties of such spaces are not always that easy to comprehend.
§1. Separation axioms in terms of nearly open sets:
The separation axioms enable us to assert with precision whether a topological space has sufficient
number of open sets as well as nearly open sets to serve the purpose that the larger the number of open sets as
well as nearly open sets, the greater is the supply of the continuous or respective continuous functions because
the concept of continuity or respective continuity is fundamental in analysis & topology and intimately linked
with open or nearly open sets.
This section highlights the overview of the separation axioms in terms of nearly open sets viz p-open,s-
open ,α- open & β-open sets.
℘ -Tk Topological Spaces (℘ = p, s, α & β; k = 0,1,2):
The literature survey on all℘ -Tk spaces (℘ = p, s, α & β; k = 0,1,2) has been brought under a
common frame work.
Definition (1.1): A topological space (X,T) is said to be :
(i) ℘ - T0 space if for each pair of distinct points x and y of X, there exists a ℘-open set A such that x∈ A but
y∉ B & that y∈ A but x∉ B.
Or
℘- T0 space if for any two distinct points x and y of X, there exists a ℘-open set containing one of them but not
the other.
(ii) ℘-T1 space if for each pair of distinct points x and y of X, there exists a pair of ℘-open sets A & B such
that x∈ A but y∉ A & that y∈ B but x∉ B.
Or
℘- T1 space if for any pair of distinct points x and y of X, there exist ℘ -open sets A & B in the manner that A
contains x but not y and B contains y but not x.
(iii) ℘- T2 space if for each pair of distinct points x and y in X, there exist two disjoint ℘-open sets A and B
such that x∈ A & y∈ B.
Or
℘- T2 space if for any two distinct points x and y of X, there exist a ℘-open sets A& B such that x∈ A, y∈ B
and A∩B = φ.
Remark (1.1): If a space (X,T) is ℘ -Tk , then it is ℘ -Tk-1 , k = 1,2. But the converse is not true.
Example (1.1): Every℘ -T0 space is not necessarily a ℘-T1 space.
Let us consider the set N of all natural numbers. Let T = { φ, N & Gn ={1,2,3,…..,n},n∈ N}.
Then (N,T) is a topological space. Obviously, every Gn is a℘ -open set where ℘ = p, s, α & β.
Clearly, the space (N,T) is a℘ -T0 space, because if we consider two distinct points m and n (m< n) then Gm =
{1,2,3,……m} is a℘ -open set containing m but not containing n and hence it is a ℘-T0 space,.
But it is not a ℘ − T1 space because if we choose Gn = {1,2,3,……,n}, then m ∈ Gm but n ∉ Gm and n ∈Gn but
m ∈Gn as m<n.
Hence,(N,T) is not a ℘-T1 space, even though it is a ℘-T0 space.
Example (1.2): Every℘ -T1 space is not necessarily a ℘ -T2 space.
Let T be the co-finite topology on an infinite set X, then (X,T) is a cofinite topological space.
3. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 99 | P a g e
Obviously, every member of T is a ℘ -open set where ℘ = p, s, α & β. Clearly, the space (X,T) is a ℘ -T1
space because if we consider two distinct points x & y of X, then {x}& {y} are finite sets and hence X –{x}, X
–{y} are members of T i.e. X-{x}& X-{y} are ℘ -open sets such that y ∈ X –{x} & x ∈ X –{y} but x∉ X-{x}&
y∉ X-{y}.
But in this case the topological space(X,T) is not ℘ -T2 space.
If possible, let (X,T) be a ℘-T2 space so that for distinct points x,y there exist ℘-open sets G & H
containing x & y respectively in the manner that G ∩ H = φ. Consequently (G ∩ H)c
= φc
i.e. Gc
∪ Hc
= X.
Now, G & H being ℘ -open sets , therefore Gc
& Hc
are both finite by definition of co-finite topology and
hence , there union X is also finite . But this contradicts the hypothesis that X is infinite which arises due to our
assumption that (X,T) is a ℘-T2 space. Hence, (X,T) is not ℘-T2 space.
Example (1.3):
(i) We consider the topological space (X,T) where X = {a,b,c,d} And
T = {φ,{a},{a,b},{c,d},{a,c,d}, X}
Here closed sets are : φ,{b},{a,b},{c,d},{b,c,d}, X.
Simple computations show that
PO(X,T) = {Φ,{a},{c},{d},{a,b},{a,c},{a,d},{c,d},{a,b,c}{a,b,d},{a,c,d}, X}.
SO(X,T) ={ Φ,{a},{a,b},{c,d},{a,c,d},X} = T.
αO(X,T) = T & βO(X,T) = PO(X,T).
Thus (X,T) is p-T0 & β-T0 space but neither p-Tk nor β- Tk space where k = 1,2.
Also ,(X,T) is not a ℘ -Tk space where ℘ = s & α; k = 0,1,2.
(ii) Let the topological space (X,T) be given by X = {a,b,c,d} And
T = {φ,{b},{c},{b,c}, X}
Simple computations show that
PO(X,T) = {Φ,{b},{c},{b,c},{a,b,c},{b,c,d}, X}.
SO(X,T)={Φ,{b},{c},{a,b},{a,c},{b,d},{b,c},{c,d},{a,b,c}{a,b,d},{b,c,d},{a,c,d},X}
αO(X,T) = PO(X,T) & βO(X,T) = SO(X,T).
Here (X,T) is s-T0, s- T1, s-T2 space as well as β-T0,β-T1 & β-T2 space. But (X,T) is neither
p-Tk nor α-Tk space where k = 0,1,2.
(iii) Let the topological space (X,T) be illustrated as:
X = {a,b,c,d} and T = {φ,{a},{b},{c},{a,b},{b,c},{c,a},{a,b,c}, X}
Simple computation provides that
PO(X,T) = {Φ,{a},{b},{c},{a,b},{b,c},{c,a},{a,b,c},X}.
SO(X,T) =P(X)-{d}.
αO(X,T) = T
& βO(X,T) = P(X)-{d}.
Here (X,T) is s-T2 space as well as β-T2 space. But it is not even p-T2,α-T2 space or p-T1, α-T1 space. Clearly,
(X,T) is a ℘-T0 space where ℘ = p, s, α & β .
Observations:
(a) In the example (i), (X,T) is not a T0 -space and also in the example (ii), (X,T) is not a T0 -space. But in the
example (iii) (X,T) is a T0 –space.
(b) Since, every open set is p-open, s-open ,α –open & β-open , hence
(b1) (X,T) is T0 –space ⟹ (X,T) is℘- T0 –space.
(b2) (X,T) is T1 –space ⟹ (X,T) is ℘-T1 –space.
(b3) (X,T) is T2 –space ⟹ (X,T) is ℘- T2 –space, where ℘ = p, s, α & β .
However, the converse of these results may not be true.
(c) These facts establish that the concepts of ℘-Tk spaces are different from the concepts of Tk -spaces where k
= 0,1,2 & ℘ = p, s, α & β .
Theorem (1.1):
A topological space (X,T) is a ℘ -T0-space iff for each pair of distinct points x &y of X , the ℘ -
cl{x}≠ ℘ -cl{y} where ℘ = p, s, α & β.
4. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 100 | P a g e
Proof:
Necessity: Let (X,T) be a ℘ -T0-space where ℘ = p, s, α & β and let x,y be any two distinct point of X. Then
we have to show that℘ -cl{x} ≠ ℘ -cl{y}. Since the space is ℘-T0, there exists a℘ -open set G containing one
of them, say x, but not containing y. Then X-G is a℘ -closed set which does not contain x but contains y. By
definition℘ -cl{y} is the intersection of all ℘-closed set containing {y}. It follows that ℘ −cl{y}⊂ X – G.
Hence x ∉ X-G implies that x∉ ℘-cl{y}. Thus x∈ ℘cl{x} but x∉ ℘ −cl{y}.
It follows that ℘ -cl{x} ≠ ℘ -cl{y}.
Sufficiency:
Let x ≠ y ⇒ ℘ − cl*x+ ≠ ℘ − cl{y} where x,y are points of X. Since ℘ -cl{x} ≠ ℘-cl{y}, there exists at
least one point z of X which belongs to one of them, say℘ -cl{x} and does not belong to ℘ −cl{y}. We claim
that x ∉ ℘-cl{y}. For if Let x∈ ℘-cl{y}.
Then ℘-cl{x} ⊆ ℘-cl{y} and so x∈ ℘-cl{x} ⊂ ℘-cl{y} which is a contradiction. Accordingly x∉ ℘-cl{y} and
consequently x ∈X –(℘-cl{y}) which is ℘ -open .
Hence, X –(℘-cl{y}) is a ℘-open set containing x but not y . It follows that (X,T) is a℘-T0-space where ℘ =
p, s, α & β.
Theorem (1.2): A topological space (X,T) is a ℘-T1- space if and only if every singleton subset {x} of X is℘ -
closed where ℘ = p, s, α & β.
Proof: The ‘if part’: let every singleton subset {x} of X be℘ -closed. We have to show that the space is ℘-T1.
Let x,y, be the two distinct point of X. Then X-{x} is a℘ -open set which contains y but does not contain x.
Similarly X-{y} is a℘ - open set which contains x but does not contain y . Hence, the space (X,T) is ℘-T1 where
℘ = p, s, α & β.
The ‘only if ‘ part : Let the space be ℘-T1 and let x be any point of X. we want to show that {x} is ℘-closed,
that is , to show that X-{x} is℘ -open. Let y∈ X-{x}.Then y≠ x. Since X is℘ -T1, there exists a ℘-open set Gy
such that y∈ Gy but x∉ Gy. It follows that y∈Gy⊂X-{x}.
Hence, X-{x} = ∪{Gy : y∈Gy} = A ℘-open set . i.e. {x} is a ℘-closed set where ℘ = p, s, α & β.
Theorem (1.3): A space (X,T) is ℘-T2 space iff for each point x∈X , the intersection of all
℘-closed set containing x is the singleton set {x}, where ℘ = p, s, α & β.
Proof: Necessity:
Suppose that (X,T) is a ℘-T2 space where ℘ = p, s, α & β. Then there exist a pair of ℘-open sets G &H
for each pair of distinct points x,y in X such that x∈G , y∈H and G ∩ H = φ. Now, G ∩ H = φ ⇒G ⊂Hc
. Hence
, x∈G⊂Hc
so that Hc
is a ℘-closed set containing x, which does not contain y as y∈H . therefore y cannot be
contained in the intersection of all ℘-closed sets which contains x. Since, y ≠ x is arbitrary, it follows that the
intersection of all ℘-closed sets containing x is the singleton set {x}. Consequently,
∩ {F: x∈ F⋀ F is ℘-closed} = {x}.
Sufficiency:
Suppose that {x} is the intersection of all ℘-closed subsets of (X,T) containing x where x is an arbitrary
point of X.
Let y be any other point of X which is different from x. Obviously, by hypothesis y does not belong to the
intersection of all ℘-closed subsets containing x. So there must exist a ℘-closed set , say N, containing x such
that y ∉N. Now, N being a ℘-closed nbd of x, there must exist a ℘-open set G such that x∈ G ⊂ N.
Thus, G and Nc
are ℘-open sets such that x∈ G, y ∈ Nc
and G ∩ Nc
= φ.Consequently(X,T) is a ℘ - T2 space
where ℘ = p, s, α & β.
Hence, the theorem.
Remark (1.2): The following example is cited in the support of above three theorems.
Let X = {a,b,c,d,e}. T = {φ,{a},{b},{a,b},{c,d},{a,c,d},{b,c,d},(a,b,c,d},{b,c,d,e},X}.
& Tc
= { φ,{a},{e},{a,e},{b,e},{a,b,e},{c,d,e},{a,c,d,e},{b,c,d,e},X}.
ThenβO(X,T)={φ,{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{b,e},{c,d},{c,e},{d,e},
{a,b,c},{a,b,d},{a,b,e},{a,c,d},{a,c,e},{b,c,d},{b,c,e},{c,d,e},{a,d,e},{b,d,e},{a,b,c,d},
{a,b,c,e},{b,c,d,e},{a,b,d,e},{a,c,d,e}, X}. = P(X)-{{e},{a,e}}.
5. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 101 | P a g e
βC(X,T) = P(X) – {{b,c,d},{a,b,c,d}}.
Here, (X,T) is a β-T2 space, consequently, it is also a β-T1 and β-T0 space.
Obviously,
(1) β-cl{x} ≠β-cl{y}; ∀ x,y∈ X& x≠ y.
(2) {a},{b},{c},{d},{e} are β-closed sets i.e. every singleton set is β-closed.
(3) ∩ {F: F∈ βC(X,T) such that x∈F} = {x}∀ x∈ X.
Definition (1.2): ℘-open mappings:
A mapping f(X,T)→(Y,σ) from one space (X,T) to another space (Y,σ) is called ℘ −open mapping (i.e. ℘ =
p,s,α,β) iff f sends ℘-open sets (i.e. ℘ = p,s,α,β –open) of (X,T) into ℘ -open sets (i.e. ℘ = p,s,α,β-open) of
(Y,σ).
Theorem (1.4):The property of a space being a ℘ - T0 space is a topological property where ℘ = p,s,α&β.
Proof:
Let f(X,T)→(Y,σ) be a one-one onto & ℘ −open mapping from a ℘ - T0 space (X,T) to any other
topological space (Y,σ). It will be established that (Y,σ) is also a ℘ - T0 space where ℘ = p,s,α,β.
Let y1 & y2 be any two distinct points of Y and as f is one-one & onto, there must exist distinct points x1 & x2 of
X such that f(x1) = y1 & f(x2) = y2……………(1)
Since, (X,T) is a ℘ - T0 space so there exists a T−℘ −open set G in manner that x1∈G but x2∉G.
Again, f, being ℘ −open, provides that f(G) is a σ−℘ −open and containing f(x1) = y1 and not containing
f(x2) = y2.
Thus, there exists a σ−℘ −open set f(G) which contains y1 and does not contain y2 and in turn (Y,σ) is a ℘ - T0
space
Again, as the property of being ℘ − T0 space is preserved under one-one , onto & ℘ −open mapping, so it
is a topological property.
Hence, the theorem.
Theorem (1.5): The property of a space being a ℘ - T1 space is a topological property where ℘ = p, s, α & β.
Proof:
Let (X,T) be a ℘ - T1 space and (Y,σ) be any other topological space such that f(X,T)→(Y,σ) is one-one
onto & ℘ −open mapping from (X,T) to (Y,σ).
It is required to prove that (Y,σ) is also a ℘ - T1 space where ℘ = p,s,α,β.
Let y1 & y2 be any two distinct points of Y and as f is one-one & onto, there must exist distinct points x1 &
x2 of X such that f(x1) = y1 & f(x2) = y2 ……………(1)
Since, (X,T) is a ℘ - T1 space so there exists a T−℘ −open sets G & H in manner that x1∈G , x2∉G & x2∈H
, x1∉H
……………………..(2)
Again, f, being ℘-open, provides that f(G) & f(H) are σ−℘ −open sets such that
f(x1) = y1 ∈ f(G) but f(x2) = y2 ∉ f(G).
& f(x2) = y2 ∈ f(H) but f(x1) = y1 ∉ f(H).
Above relations show that (Y,σ) is also a ℘ - T1 space.
Again, as the property of being ℘ - T1 space is preserved under one-one , onto & ℘ −open mapping, so it is
a topological property.
Hence, the theorem.
Theorem (1.6): The property of a space being a ℘ - T2 space is a topological property where ℘ = p,s,α & β.
Proof: Let (X,T) be a ℘ −T2 space and (Y,σ) be any other topological space such that f(X,T)→(Y,σ) is one-
one onto & ℘ −open mapping from (X,T) to (Y,σ).
It is required to prove that (Y,σ) is also a ℘ - T2 space where ℘ = p,s,α &β.
Let y1 & y2 be any two distinct points of Y and as f is one-one & onto, there must exist distinct points x1 &
x2 of X such that f(x1) = y1 & f(x2) = y2 ……………(1)
Since, (X,T) is a ℘ - T2 space so there exists a T−℘ −open set G & H such that x1∈G , x2∉G & G ∩ H = φ
………………..(2)
Again, f, being ℘ −open, provides that f(G) & f(H) are σ−℘ −open sets such that
f(x1) = y1 ∈ f(G) but f(x2) = y2 ∈ f(H) and
& G ∩ H = φ ⇒ f(G ∩ H) = φ ⇒ f( G) ∩ f(H )= φ.
Above relations show that (Y,σ) is also a ℘ - T2 space.
6. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 102 | P a g e
Again, as the property of being ℘ - T2 space is preserved under one-one , onto & ℘-open mapping, so it is a
topological property.
Hence, the theorem.
§2. ℘ - R0 spaces where ℘ = p,s,α & β.
In this section, the notion of ℘ - R0 spaces where ℘ stands for p,s,α,β is introduced and some basic
properties are discussed. But before we take up it, we project the notion of the ℘ -kernel of a set A of a space
(X,T) and the ℘ -kernel of a point x of a space (X,T) in the following manner:
Definition (2.1):
𝐼𝑛 𝑎 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑠𝑝𝑎𝑐𝑒 (𝑋, 𝑇), 𝑖𝑓 𝐴
⊆ 𝑋, 𝑡𝑒𝑛 𝑡𝑒 ℘ − 𝑘𝑒𝑟𝑛𝑒𝑙 𝑜𝑓 𝐴, 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 ℘ − 𝑘𝑒𝑟(𝐴), 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑡𝑜 𝑏𝑒 𝑡𝑒 𝑠𝑒𝑡 ℘
− 𝑘𝑒𝑟(𝐴) = ∩ { 𝒪 ∈ ℘𝑂(𝑋, 𝑇)| 𝐴 ⊆ 𝒪}.
Definition (2.2):
If x be a point of a topological space (X,T), then the ℘ -kernel of x , denoted by ℘ -ker({x}) is defined to be
the set ℘ − ker({x})= ∩{ 𝓞 ∈ ℘𝑂(X,T)| x∈ 𝓞 }.
Lemma (2.1):
If A be a subset of of a topological space (X,T), then ℘ −ker(A) = ∩{x∈ X |℘ -cl({x})∩ 𝐴 ≠ 𝛷}.
Proof: Let x∈ ℘ −ker(A) where A⊆ X & (X,T) is a topological space. On the contrary, we assume that ℘ -
cl({x})∩ A = Φ. Hence ,x ∉ X – {℘ -cl({x})} which is a ℘-open set containing A. This is impossible as x
∈ ℘ −ker(A). Consequently, ℘ -cl({x})∩ A≠ Φ.
Again let , ℘ -cl({x})∩ ≠ Φ exist and at the same time let x ∉ ℘ −ker(A). This means that there exists a ℘-
open set B containing A and x ∉ B.
Let y ∈ ℘ -cl({x})∩ A. therefore, B is a ℘ −nbhd of y for which x ∉ B. By this contradiction, we have x ∈ ℘-
ker(A).
Hence, p-ker({A})=∩ {x∈ X |℘ -cl({x})∩ A ≠ Φ}.
Definition (2.3): ℘ − R0 spaces:
A topological space (X,T) is said to be a ℘ − R0 space if every ℘ − open set contains the ℘ − closure of
each of its singletons, where ℘ = p,s,α & β.
The implications between ℘ − R0 spaces are indicated by the following diagram:
R0 space ⟹ α-R0 space ⟹ s-R0 space
⇓ ⇓
p- R0 space β- R0 space.
We, however, know that a R0-space is a topological space in which the closure of the singleton of every point of
an open set is contained in that set.
None of the above implications in the diagram is reversible, as illustrated by the following examples:
Example (2.1):
Let X = {a,b,c}, T = {φ,(a,b},X}. Then PO(X,T) = {φ,{a},{b},{a,b},{b,c},{c,a},X}.
& PC(X,T) = {φ,{b},{a},{c},{a,c},{b,c},X}.
Hence, (X,T) is a p- R0 space.
Again, αO(X,T) = {φ, {a,b},X} = sO(X,T).
& αC(X,T) = {φ,{c},X} = sC(X,T).
Since, α-cl({a}) = X ⊄{a,b}∈ αO(X,T), hence,(X,T) is not a α-R0 space.
Similar is the reason for (X,T) to be not a s-R0 space.
Example (2.2):
Let X = {a,b,c} , T = {φ,{a},{b},{a,b},X}.
Then sO(X,T) = {φ,{a},{b},{a,b},{b,c},{c,a},X} = βO(X,T).
& sC(X,T) = {φ,{b},{a},{c},{a,c},{b,c},X} = βC(X,T).
Hence, (X,T) is a s-R0 space as well as β-R0 space.
Again, PO(X,T) = {φ,{a},{b},{a,b},X} = αO(X,T).
& PC(X,T) ={φ,{c},{a,c},{b,c},X} = αC(X,T).
7. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 103 | P a g e
Since, pcl({a}) = {a,c}⊄{a,b}∈ PO(X,T), hence, (X,T) is not a p-R0 space.
Similar is the reason for (X,T) to be not an α-R0 space.
Remark (2.1):
(1) The concepts of p-R0 space and s-R0 are independent. Example (2.1) shows that the space (X,T)is p-R0 but
s-R0 where as in example (2.2), the space (X,T) is s-R0 but not p-R0.
(2) The notion of α-R0 does not imply the notion of R0 as it is shown by the following example.
Example (2.3):
Let X be an infinite set and p∈X be a fixed point. Let T = {φ &G ⊂ X –{p}& Gc
is finite}.
It can be observed that if G is an open set and x∈ G, then cl({x}) = X ⊄G. So, (X,T) is not a R0 space but as X is
α- T1 so every {x} is α-closed so α-cl({x}) = {x} ⊂G, ∀ x ∈ G & G ∈ αO(X,T). Hence, (X,T) is an α-R0 space.
Example(2.4):
Let X = {a,b,c,d,e}. T = {φ,{a},{b},{a,b},{c,d},{a,c,d},{b,c,d},(a,b,c,d},{b,c,d,e},X}.
& Tc
= { φ,{a},{e},{a,e},{b,e},{a,b,e},{c,d,e},{a,c,d,e},{b,c,d,e},X}. Then
PO(X,T)={φ,{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},
{a,b,d},{a,c,d},{b,c,d},{b,c,e},{a,b,c,d},{a,b,c,e},{b,c,d,e},{a,b,d,e}X}.
PC(X,T)={φ,{a},{c},{d},{e},{a,d},{a,e},{b,e},{c,e},{d,e},{a,b,e},{a,c,e},{a,d,e},{b,c,e},{b,d,e},
{c,d,e},{a,b,c,e},{a,b,d,e},{a,c,d,e},{b,c,d,e},X}.
Since,p-cl({b})= {b,e} ⊄ {a,b,c,d}∈ PO(X,T), Hence, (X,T) is not a p-R0 space.
Again, βO(X,T) = P(X)-{{e},{a,e}} & βC(X,T) = P(X) – {{b,c,d},{a,b,c,d}}.
Since, (X,T) is a β-T1 space so every {x} is β-closed which means that β-cl({x}) = {x} ⊂G∀ x ∈ G &
G∈βO(X,T). Consequently, (X,T) is a β-R0 space.
Next,SO(X,T)={φ,X,{a},{b},{a,b},{b,e},{c,d},{a,b,e},{a,c,d},{b,c,d},{c,d,e},{a,b,c,d},{b,c,d,e},{a,c,d,e}}.
SC(X,T)={φ,X,{a},{b},{e},{a,b},{a,e},{b,e},{c,d},{a,b,e},{a,c,d},{c,d,e},{a,c,d,e},{ b,c,d,e}}.
Here, (X,T) is a s-R0 space.
Also, αO(X,T) = T & αC(X,T) = { φ,X,{a},{e},{a,e},{b,e},{a,b,e},{c,d,e},{a,c,d,e},{b,c,d,e}}.
Here, (X,T) is not an α-R0 space.
We, now, mention the following lemmas with proofs, useful in the sequel.
Lemma (2.2): In a topological space (X,T), for each pair of distinct points
x,y, ∈ X, x ∈ ℘ −cl({y}) ⇔ y ∈ ℘ −ker({x}), where ℘ = p,s, 𝛼 & β.
Proof:
Suppose that y∉ ℘ −ker({x}). Then there exists a ℘ -open set V containing x such that y ∈ V. Therefore,
we have x∉ ℘ −cl({y}).
This means that 𝑦 ∉ ℘ − 𝑘𝑒𝑟({𝑥}) ⇒ 𝑥 ∉ ℘ − 𝑐𝑙({𝑦}).
𝑖. 𝑒. ℸ 𝑥 ∉ ℘ − 𝑐𝑙({𝑦}) ⇒ ℸ 𝑦 ∉ ℘ − 𝑘𝑒𝑟({𝑥})
𝑖. 𝑒. 𝑥 ∈ ℘ − 𝑐𝑙({𝑦}) ⇒ 𝑦 ∈ ℘ − 𝑘𝑒𝑟({𝑥}).
Similar is the argument for the proof of the converse i.e.
y ∈ ℘ −ker({x})⇒ x∈ ℘ −cl({y}).
Hence, the theorem.
Lemma (2.3): The following statement are equivalent for each pair of points x & y in a topological space
(X,T):
(a) ℘ − 𝑘𝑒𝑟({𝑥}) ≠ ℘ − 𝑘𝑒𝑟({𝑦}).
(b) ℘ − 𝑐𝑙({𝑥}) ≠ ℘ − 𝑐𝑙({𝑦}). 𝑊𝑒𝑟𝑒 ℘ = 𝑝, 𝑠, 𝛼 & 𝛽.
Proof: (a) ⇒(b):
𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑡𝑎𝑡 ℘ − 𝑘𝑒𝑟({𝑥}) ≠ ℘ − 𝑘𝑒𝑟({𝑦}), 𝑡𝑒𝑛 𝑡𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑧 𝑖𝑛 𝑋 𝑠𝑢𝑐 𝑡𝑎𝑡 𝑧 ∈ ℘ −
𝑘𝑒𝑟({𝑥}) 𝑎𝑛𝑑 𝑧 ∉ ℘ − 𝑘𝑒𝑟({𝑦}).
Since 𝑧 ∈ ℘ − 𝑘𝑒𝑟({𝑥}), 𝑒𝑛𝑐𝑒 𝑥 ∈ ℘ − 𝑐𝑙({𝑧}). This means that
{𝑥} ∩ ∈ ℘ − 𝑐l({z}) ≠ φ.
𝐵𝑦 𝑧 ∉ ℘ − 𝑘𝑒𝑟({𝑦}), 𝑤𝑒 𝑎𝑣𝑒 {𝑦} ∩ ℘ − 𝑐𝑙({𝑧}) = φ.
Since, x ∈ ℘-cl({z}), ℘-cl({x}) ⊂ ℘-cl({z}) and {y}∩ ℘-cl(*x+) = φ.
Hence, ℘-cl({x}) ≠ ℘-cl({y}).
8. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 104 | P a g e
(b) ⇒(a): Suppose that ℘-cl({x}) ≠ ℘-cl({y}). Then there exists a point z in X such that z ∈ ℘-cl({x}) and
𝑧 ∉ ℘ −cl({y}).There, there exists a ℘ -open set containing z and therefore x but not y i.e. y ∉ ℘-ker({x}).
Hence, ℘ −ker({x}) ≠ ℘ −ker({y}).
Hence, the theorem.
Theorem (2.1):
A space (X,T) is℘ -R0 space if and only if for each pair x,y of distinct points in X,
℘-cl({x}) ∩ ℘-cl(*y+) = φ or *x,y+ ⊂ ℘-cl({x}) ∩ ℘-cl({y}) where ℘ = p,s, α & β.
Proof: Necessity :
Let (X,T) be a℘ -R0 space and x,y ∈X, x ≠y. On the contrary, suppose that
℘-cl({x}) ∩ ℘-cl({y}) ≠ φ & *x,y+ ⊄ ℘-cl({x}) ∩ ℘-cl({y}) .
Let z ∈ ℘-cl({x}) ∩ ℘-cl({y}) & x ∉ ℘-cl({x}) ∩ ℘-cl({y}).
Then x ∉ ℘-cl({y}) and x ∈(p-cl{y})c which is a ℘ -open set. But ℘-cl({x}) ⊄ [℘-cl({y})]c.
which appears as a contradiction as (X,T) is a ℘-R0 space.
Hence, for each pair of distinct points x,y of X, we have ℘-cl({x}) ∩ ℘-cl(*y+) = φ or
{x,y}⊂ ℘-cl({x}) ∩ ℘-cl({y}) .
Sufficiency :
Let U be a ℘ -open set and x ∈U. Suppose that ℘-cl({x}) ⊄ U. So there is a point
y∈ ℘-cl({x}) such that y ∉ U and ℘-cl({y}) ∩ U = φ.
Since, Uc is ℘ -closed & y ∈ Uc, hence, {x,y} ⊄ ℘-cl({y}) ∩ ℘-cl({x}) and thus
℘-cl({x}) ∩ ℘-cl({y}) ≠ φ . Consequently, the assumption of the condition provides that (X,T) is ℘- R0 space.
Hence, the theorem.
Theorem (2.2): For a topological space (X,T), the following properties are equivalent :
(a) (𝑋, 𝑇) 𝑖𝑠 𝑎℘ − 𝑅0 𝑠𝑝𝑎𝑐𝑒;
(b) ℘ − 𝑐𝑙({𝑥}) = ℘ − 𝑘𝑒𝑟({𝑥}), ∀ 𝑥 ∈ 𝑋, 𝑤𝑒𝑟𝑒 ℘ = 𝑝, 𝑠, 𝛼 & 𝛽.
Proof: (a) ⇒(b):
Let (X,T) be a℘ − 𝑅0 𝑠𝑝𝑎𝑐𝑒.
𝐵𝑦 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 (2.2), 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑥 ∈ 𝑋, 𝑤𝑒 𝑎𝑣𝑒 ℘ − 𝑘𝑒𝑟({𝑥}) = ∩ { 𝒪 ∈ ℘𝑂(𝑋, 𝑇)| 𝑥 ∈ 𝒪}.
And by definition (2.3) , each ℘ -open set θ containing x contains ℘ −cl({x}).
Hence, ℘ − 𝑐𝑙({𝑥}) ⊂ ℘ − 𝑘𝑒𝑟({𝑥}).
𝐿𝑒𝑡 𝑦 ∈ ℘ − 𝑘𝑒𝑟({𝑥}), 𝑡𝑒𝑛 𝑥 ∈ ℘ − 𝑘𝑒𝑟({𝑦}) 𝑏𝑦 𝑙𝑒𝑚𝑚𝑎(2.2), 𝑎𝑛𝑑 𝑠𝑜 ℘ − 𝑐𝑙({𝑥})
= ℘ − 𝑐𝑙({𝑦}). 𝑇𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑦 ∈ ℘ − 𝑐𝑙({𝑥}). 𝑇𝑒𝑠𝑒 𝑚𝑒𝑎𝑛 𝑡𝑎𝑡 ℘ − 𝑘𝑒𝑟({𝑥})
⊂ ℘ − 𝑐𝑙({𝑥}).
𝐻𝑒𝑛𝑐𝑒, ℘ − 𝑐𝑙({𝑥}) = ℘ − 𝑘𝑒𝑟({𝑥}).
(b)⇒(a):
Suppose that 𝑓𝑜𝑟 a topological space (𝑋, 𝑇), ℘ − 𝑐𝑙({𝑥}) = ℘ − 𝑘𝑒𝑟({𝑥}) ∀ 𝑥 ∈ 𝑋.
Let G be any ℘ −open set in (X,T) , then for every 𝑝 ∈ 𝐺, ℘ − 𝑘𝑒𝑟({𝑝}) = ∩ { 𝐺 ∈ ℘𝑂(𝑋, 𝑇)| 𝑝 ∈
𝐺}. 𝐵𝑢𝑡 ℘ − 𝑐𝑙({𝑝}) = ℘ − 𝑘𝑒𝑟({𝑝}) by hypothesis. Hence, combing these two, we observe that for every
p∈G∈ ℘O(X,T), ℘-cl({x})∈G. Consequently, (X,T) is a ℘- R0 space.
Hence, the theorem.
Theorem (2.3): For a topological space (X,T) , the following properties are equivalent:
(a) (X,T) is a ℘ −R0 space.
(b) If F is ℘-closed, then F = ℘- ker (F);
(c) If F is ℘-closed and x ∈ F, then ℘- ker ({x}) ⊂ F.
(d) If x∈ X, then ℘-ker({x}) ⊂ ℘-cl({x}).
Proof: (a) ⇒(b):
9. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 105 | P a g e
(a) Let (X,T) is a ℘ −R0 space & F, a ℘-closed set. Let x ∉ F, then Fc is a ℘ -open set containing x, so that
℘-cl({x}) ⊂ Fc as (X,T) is a ℘ −R0. This means that ℘-cl({x})∩ F= φ and by lemma (2.1), x ∉ ℘- ker
(F).
Therefore, ℘- ker (F) = F.
(b) ⇒(c):
In general, A ⊂ B ⇒ ℘- ker (A) ⊂ ℘- ker (B), Hence, it follows that for x ∈ F, {x} ⊂ F ⇒ ℘- ker({x}) ⊂
℘- ker(F) = F as F is ℘-closed.
(c) ⇒(d):
Since, x ∈ ℘-cl({x}) and ℘-cl({x}) is ℘ -closed, hence, using (c) we get
℘- ker({x}) ⊂ ℘- cl({x}).
(d)⇒(a) :
Let (X,T) be a topological space in which ℘- ker({x}) ⊂ ℘- cl({x}) for every x ∈ X.
Let y ∈ ℘-cl({x}), then x ∈ ℘-ker({y}), since, y ∈ ℘-cl({y}) and ℘-cl({y}) is ℘-closed, by hypothesis x ∈
℘-ker({y})⊂ ℘-cl({y}). Therefore y ∈ ℘-cl({x}) ⇒ x ∈ ℘-cl({y}). Similarly, x ∈ ℘-cl({y}) implies y ∈ ℘-
cl({x}). Thus (X,T) is ℘-R0 space, using theorem (2.4).
Hence, the theorem.
Theorem (2.4): for a topological space (X,T) , the following properties are equivalent:
(a) (𝑋, 𝑇) 𝑖𝑠 𝑎 ℘ − 𝑅0 𝑠𝑝𝑎𝑐𝑒;
(b) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑝𝑜𝑖𝑛𝑡𝑠 𝑥 & 𝑦 𝑜𝑓 𝑋, 𝑥 ∈ ℘ − 𝑐𝑙({𝑦}) ⟺ 𝑦 ∈ ℘ − 𝑐𝑙({𝑥}).
Proof: (a) ⇒(b):
Let (X,T) be a ∈ ℘ −R0 space. Let x & y be any two points of X. Assume that x ∈ ℘ −cl({y}) and D is any
℘ −open set such that y∈ D.
Now, by hypothesis, x ∈ D. Therefore, every ℘ −open set containing y contains x. Hence, y ∈ ℘ −cl({x}) i.e.
x ∈ ℘ − 𝑐𝑙({𝑦} ⇒ 𝑦 ∈ ℘ − 𝑐𝑙({𝑥}). The converse is obvious and x∈ ℘ − 𝑐𝑙({𝑦}) ⟺ 𝑦 ∈ ℘ − 𝑐𝑙({𝑥}).
(b) ⇒(a):
Let U be ℘-open set and x ∈ U. If y ∉ U, then x ∉ ℘-cl({y}) and hence, y ∉ ℘ −cl({x}). This implies that
℘ −cl({x}) ⊂ U. Hence, (X,T) is a ℘-R0 space.
Hence, the theorem.
§𝟑. ℘ − 𝑹𝟏 𝒔𝒑𝒂𝒄𝒆𝒔 𝒘𝒉𝒆𝒓𝒆 ℘ = 𝒑, 𝒔, 𝜶 & 𝛽.
This section includes the notion of ℘ −R1 spaces where ℘ stands for p,s, 𝛼 & β and their basic properties.
Definition (3.1):
A topological space (X,T) is said to be a ℘-R1 space if for each pair of distinct points x & y of X with ℘-cl
({𝑥}) ≠
℘ − 𝑐𝑙({𝑦}) , 𝑡𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑑𝑖𝑠𝑗𝑜𝑖𝑛𝑡 𝑝𝑎𝑖𝑟 𝑜𝑓 ℘ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡𝑠 𝑈 𝑎𝑛𝑑 𝑉 𝑠𝑢𝑐 𝑡𝑎𝑡 ℘ − 𝑐𝑙 ({𝑥}) 𝑈 & ℘ −
𝑐𝑙 ({𝑦}) ⊂ 𝑉, 𝑤𝑒𝑟𝑒 ℘ = 𝑝, 𝑠, 𝛼 & 𝛽.
Theorem (3.1): If (X,T) is a ℘-R1 space, then it is a ℘-R0 space.
Proof: Suppose that (X,T) is a ℘-R1 space where ℘ = p,s, α & β.
Let U be a ℘-open set and x ∈U. then for each point y∈Uc, ℘-cl ({x}) ≠ ℘-cl({y}).
Since,(X,T) is a ℘-R1 space, there exist a pair of ℘-open sets Uy & Vy such that ℘-cl ({x}) ⊂ Uy & ℘-cl ({y})
⊂ Vy & Uy ∩ Vy = φ.
Let A = ∪{Vy: y ∈Uc}. Then Uc ⊂A, x ∈ A and A is a ℘-open set.
Therefore, ℘-cl({x}) ⊂Ac ⊂ U which means that (X,T) is a ℘-R0 space.
Hence, the theorem.
Example (3.1):
If p be a fixed point of (X,T) with T as the co-finite topology on X given as
T = *φ, X,G with G ⊂ X – {p} & Gc is finite.}, then the space (X,T) is ℘-R0 but it is not ℘-R1 where ℘ = p,s, α
& β.
Theorem (3.2):
10. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 106 | P a g e
A space (X,T) is a ℘ −R1 space iff for each pair of distinct points x & y of X with ℘ −cl ({x}) ≠ ℘-cl({y}) ,
there exist disjoint pair of ℘ −open sets U and V such that x ∈U,y ∈ V & U ∩ V = φ.
Necessity:
Let (X,T) be a ℘ −R1 space. By definition (3.1), for each pair of distinct points x & y of X with ℘ −
𝑐𝑙 ({𝑥}) ≠ ℘-cl({y}) ,there can always be obtained disjoint pair of ℘-open sets U and V such 𝑡𝑎𝑡 ℘ −
𝑐𝑙 ({𝑥}) ⊂ 𝑈 & ℘ − 𝑐𝑙 ({y}) ⊂ V where U ∩ V = φ. We, however, know that 𝑝 ∈ ℘ −
𝑐𝑙 ({𝑝}), ∀ 𝑝 ∈ 𝑋. 𝐻𝑒𝑛𝑐𝑒, 𝑥 ∈ 𝑈, 𝑦 ∈ 𝑉 & 𝑈 ∩ 𝑉 = 𝜑.
Sufficiency:
Let x ,y∈ X and x ≠ y such 𝑡𝑎𝑡 ℘ − 𝑐𝑙 ({𝑥}) ≠ ℘ − 𝑐𝑙({𝑦}) . Also let U & V be 𝑑𝑖𝑠𝑗𝑜𝑖𝑛𝑡 ℘ -open sets for
which x ∈U,y ∈ V.
Since, U ∩ V = φ, hence, 𝑥 ∈ ℘ − 𝑐𝑙 ({𝑥}) ⊂ 𝑈 & 𝑦 ∈ ℘ − 𝑐𝑙 ({𝑦}) ⊂ V.Consequently, (X,T) is a ℘ −R1
space. Hence, the theorem.
Corollary (3.1):
Every ℘ −T2 space is ℘ −R1 space, but the converse is not true. However, we have the following
result.
Theorem (3.3): Every ℘ −T1 & ℘ −R1 space is ℘ −T2 space.
Proof:
Let (X,T) be a ℘ −T1 as well as ℘ −R1 space. Since, (X,T) is a ℘ −T1 space, hence, ℘ −cl ({x}) =
{x} ≠ {y} = ℘ −cl({y}) for x,y ∈ X & x ≠ y.
Now, theorem (3.2) provides that as (X,T) is a ℘ −R1 space and here, x ,y∈ X and x ≠ y such that
℘ −cl ({x}) ≠ ℘ −cl({y}), so there exist ℘ -open sets U & V such that x ∈U,y ∈ V & U ∩ V = φ.
Consequently, (X,T) is a ℘ −T2 space.
Hence, the theorem.
Theorem (3.4): For a topological space (X,T) , the following properties are equivalent:
(a) (X,T) is a ℘ −R1 space;
(b) For any two distinct points x,y ∈ X with ℘ −cl ({x}) ≠ ℘ −cl({y}), there exist ℘ -closed sets F1 & F2 such
that x ∈ F1, y∈ F2 x ∉F2, y ∉F1 and F1∪ F2 = X,where ℘ = p,s, 𝛼 & β.
Proof: (a) ⇒(b):
Suppose that (X,T) is a ℘ −R1 space. Let x,y ∈X and x ≠ y and with ℘ −cl ({x}) ≠ ℘ −cl({y}),by Theorem
(3.2), there exist ℘ -open sets U & V such that x ∈U,y ∈ V . Then, F1 = Vc
𝑖𝑠 a ℘ -closed set & F2 = Uc
is also
℘ -closed set such that x ∈ F1, y∈ F2 x ∉F2, y ∉F1 and F1∪ F2 = X.
(b)⇒(a):
Let x,y ∈ X such that ℘ −cl ({x}) ≠ ℘ −cl({y}).This means that ℘ −cl ({x}) ∩ ℘ −cl({y}) = φ.
By the assume condition (b), there exist℘ -closed sets F1 & F2 such that x ∈ F1, y∈ F2 , x ∉F2, y ∉F1 and F1∪ F2
= X.
Therefore, x ∈
c
F 2 = U = A ℘ -open set.
& y ∈
c
F 1 = V = A ℘ -open set.
Also U ∩ V = φ.
These facts indicates that
x∈ ℘-cl ({x}) ⊂ U & y∈ ℘-cl ({y}) ⊂ V such that U ∩ V = φ. Consequently, (X,T) is a
℘-R1 space.
Hence the theorem.
§4. ℘ −symmetry of A space & ℘ − generalized closed set:
We, now, define ℘ − symmetry of a space & (X,T) & ℘ −generalized closed set (briefly ℘g-closed set) in a
space (X,T) as:
Definition (4.1): A space(X,T) is said to be ℘ −symmetric if for every pair of points x,y. in X , x∈ ℘ −cl
({y})⇒ y∈ ℘ −cl ({x}) where ℘ = p,s, 𝛼 & β.
Definition (4.2): A subset A of a space (X,T) is said to be a ℘ −generalized closed set(briefly ℘g-closed set) if
℘ −cl ({A}) ⊆ U whenever A⊆U & U is ℘ −open in X
11. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 107 | P a g e
where℘ = p,s, 𝛼 & β.
Lemma (4.1): Every ℘ -closed set is a ℘g-closed set but the converse is not true where ℘ = p,s, 𝛼 & β.
Proof:
It follows from the fact that whenever A is ℘ −closed set, we have℘-cl(A) = A for ℘ = p,s, 𝛼 & β,so the
criteria ℘ −cl ({A}) ⊆ U whenever A⊆ U & U is ℘ − open exists & A turns to be a ℘g-closed set.
But the converse need not to be true as illustrated by the following example:
Let X = {a,b,c,d} And T = {φ,{a},{a,b},{c,d},{a,c,d}, X}
Here closed sets are : φ,{b},{a,b},{c,d},{b,c,d}, X.
Then , Ts ={ Φ,{a},{a,b},{c,d},{a,c,d},X}, &
C
sT = {φ,{b},{a,b},{c,d},{b,c,d},X}.
Now,
C
sgT = the class of all sg-closed sets.
= {φ,{b},{c},{d},{a,b},{c,d},{b,c},{b,d},{a,b,c},{a,b,d},{b,c,d},X}.
Therefore, {c},{d},{b,c},{b,d},{a,b,c},{a,b,d} are sg-closed sets but not s-closed.
Also, T = sT ,
C
T =
C
sT & C
gT
C
sgT which show that {c},{d},{b,c},{b,d},{a,b,c},{a,b,d} are αg-closed
sets but not α-closed sets.
Similarly, the other cases can be dealt with.
Theorem (4.1): A space (X,T) is ℘ −symmetric if and only if {x} is ℘g-closed for each x∈X, where ℘ = p,s,
𝛼 & β.
Proof: Necessity: Let (X,T) be ℘-symmetric , then for distinct points x,y of X ,
y∈ ℘ −cl ({x})⇒ x∈ ℘ −cl ({y}) where ℘ = p,s, 𝛼 & β.
Let {x}⊂D where D is 𝑎 ℘ −open set in (X,T). Let ℘ −cl ({x})⊄ D. This means that (℘-cl ({x}))∩Dc
≠φ. Let
y ∈ (℘ −cl ({x}))∩Dc
. Now, we have x∈ ℘ −cl ({y})⊂Dc
and x ∉ D. but this is a contradiction . Hence, ℘ −cl
({x})⊂D whenever {x} ⊂D & D is ℘ -open. Consequently, {x} is a ℘g-closed set.
Sufficiency: Let in a space (X,T) , each {x} is a ℘g-closed set where x∈ X. Let x,y∈ X & x ≠ y such that x∈
℘ −cl ({y}) but y∉ ℘ −cl ({x}…………(1)
This implies that y∈ ( ℘ − 𝑐𝑙 ({𝑥}))𝑐
⇒ {𝑦} ⊂ ( ℘ − 𝑐𝑙 ({𝑥}))𝑐
⇒ ℘ − 𝑐𝑙 ({𝑦}) ⊂ ( ℘ − 𝑐𝑙 ({𝑥}))c
…………………(2)
as {y} is a ℘g-closed set by the assumption.
Now, {x} ⊂ ( ℘ −cl ({x}))c
from (1) & (2). this is an assumption contradiction which arises due to the
acceptance of (1) and consequently , we have
x∈ ℘ −cl ({y})⇒ y∈ ℘ −cl ({x}) ; for every x ≠y . Therefore, the space (X,T) is ℘ −symmetric.
𝐻𝑒𝑛𝑐𝑒 𝑡𝑒 𝑡𝑒𝑜𝑟𝑒𝑚.
𝑪𝒐𝒓𝒐𝒍𝒍𝒂𝒓𝒚 (𝟒. 𝟏): 𝐼𝑓 𝑎 𝑠𝑝𝑎𝑐𝑒 (𝑋, 𝑇) 𝑖𝑠 ℘ −T1 space , then it is ℘ −symmetric, where ℘ = p,s, 𝛼 & β.
Proof: In a ℘-T1 space, singleton sets 𝑎𝑟𝑒 ℘ −closed by Theorem (1.2) , and therefore ℘g-closed by Lemma
(4.1). By Theorem (4.1), the space (X,T) is ℘ −symmetric, where ℘ = p,s, 𝛼 & β.
Remark (4.1):
The converse of the corollary (4.1) is not necessarily true as shown in the following example:
Let X = {a,b,c,d,e}. T = {φ,{a},{b},{a,b},{c,d},{a,c,d},{b,c,d},(a,b,c,d},{b,c,d,e},X}.
Then
C
sT ={φ,{a},{b},{e},{a,b},{a,e},{b,e},{c,d},{a,b,e},{a,c,d},{c,d,e},{b,c,d,e}, {a,c,d,e}, X}
sT ={φ,{b},{a},{a,b},{b,e},{c,d},{a,b,e},{a,c,d},{b,c,d},{c,d,e},{a,b,c,d},{a,c,d,e},
{b,c,d,e},X}.
The space (X,T) is not s-T1 but s-symmetric.
Theorem (4.2): for a topological space (X,T) , the following properties are equivalent :
(a) (X,T) is a℘ −symmetric & ℘ −T0 space;
(b) (X,T) is ℘ −T1 space.
12. Dr. Thakur C. K. Raman Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -6) April 2015, pp. 97-108
www.ijera.com 108 | P a g e
Proof: (a) ⇒(b):
Let x,y ∈ X and x ≠y. Since, (X,T) is a ℘ −T0 space, hence we may assume that x ∈ G1⊂{y}c
for some ℘ −
open set G1 . Then x ∉ ℘ −cl ({y}). Consequently y ∉ ℘ −cl ({x}). There exists a ℘ −open set G2 such that y
∈ G2⊂{x}c
. Therefore, (X,T) is a ℘ −T1 space.
(b) ⇒(a):
Corollary (4.1) depicts that (X,T), being ℘ −T1 space is ℘ −symmetric .
Remark (1.1) provides that (X,T) being ℘ −T1 space is necessarily ℘ −T0 space.
The above two facts together establish that (b) ⇒(a).
Hence ,the theorem.
Corollary (4.2): If (X,T) is ℘ −symmetric, then (X,T) is ℘ −T0 ⇔ (X,T) is ℘ −T1.
Proof: Here, „⇒” follows from Theorem (4.2) & „⇐‟ follows from Remark (1.1).
II. Conclusion
An overview of separation axioms by nearly open sets focuses its attention on the literature of ℘ −Tk( k=
0,1,2 & ℘ = p,s, 𝛼 & β) spaces in the compact form in this paper.
The study of ℘ −R0 & ℘ −R1 spaces has been enunciated and the related properties are kept ready at a
glance. The ℘- symmetry of a topological space along with example and basic results has been exhibited at one
place.
The future scope of the overview is to compile the literature & research concern with ℘ −T1/2 spaces(℘ =
p,s, α & β) and the related fundamental properties & results are to be prepare as a ready reckoning at a glance.
REFERENCES
[1] D. Andrijevic, “Semi-Preopen Sets”, Mat. Vesnik, 38, (1986), 24-32.
[2] D. Andrijevic, “On b -Open Sets”, Mat. Vesnik, 48, (1996), 59-64.
[3] S. Arya and T. Nour, Characterizations of s-normal spaces, Indian J. Pure Appl. Math. 21 (1990), 717–
719.
[4] N. Bourbaki, General Topology Part 1 Addison Wesley Publ. Co. Reading Mass,1966.
[5] Miguel Caldas Cueva; A Research on Characterizations of Semi-T1/2 Spaces, Divulgaciones
Matematicas Vol. 8 No. 1 (2000), pp. 43-50.
[6] E. Ekici And M. Caldas, Slightly γ- Continuous Functions, Bul. Soc. Pran. Mat (35). 22(2)(2004), 63-
74.
[7] S. N. El-Deeb, I. A. Hasanein, A. S. Mashhour and T. Noiri, On p-regular spaces, Bull. Math. Soc. Sci.
Math. R. S. Roumanie, 27(75) (1983), 311–315.[12]
[8] Y. Gnanambal,On generalized p-regular closed sets in topological spaces, Indian J. Pure Appl.Math.,
28 (1997), 351–360.
[9] D. Jankovi´c and I. Reilly, On semi-separation properties, Indian J. Pure Appl. Math. 16 (1985), 957–
964.
[10] Vidyottama Kumari & Dr. Thakur C.K.Raman, On Characterization Of b-Open Sets In Topological
Spaces., IJAIEM, Vol .2, Issue 12, Dec. 2013,229-235.
[11] Vidyottama Kumari , A Discourse On Analytical Study Of Nearly Open Sets .IOSR-JM,Volume 10,
Issue 5 Ver. Iii (Sep-Oct. 2014), Pp 30-41.
[12] N.Levine, Semi-Open Sets And Semi- Continuity In Topological Spaces, Amer.Math. Monthly
70(1963), 36-41.
[13] A. S. Mashhour, M.E. Abd- El. Monsef And S.N. El Deeb, On Pre- Continuous And Weak Pre-
Continuous Mappings, Proc. Math. And Phys. Soc. Egypt 53(1982), 47-53.
[14] O.Njasted, On Some Classes Of Nearly Open Sets, Pacific J. Math15 (1965), 961-970.
[15] Dr. Thakur C.K.Raman & Vidyottama Kumari, -Generalized &
- Separation Axioms for
Topological Spaces.IOSR-JM, May-June 2014, Vol-10, Issue 3, Ver. 6, 32-37.