This document discusses various separation axioms related to rg-open sets. It begins by defining rg-closed sets and rg-limit points. It then introduces concepts such as rg-normal, rg-US, and rg-S1 spaces. The main part of the document characterizes properties of rg-T0 spaces and rg-R0 spaces. It shows several equivalent definitions for these spaces and establishes various properties that hold in such spaces, such as every rg-limit point being a rg-T0-limit point. It also discusses rg-R1 spaces and shows properties that hold in rg-compact rg-R1 spaces, such as being a Baire space.
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.
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 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.
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.
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 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.
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
Tim Maudlin: New Foundations for Physical GeometryArun Gupta
New Foundations for Physical Geometry
Original URL: http://www.unil.ch/webdav/site/philo/shared/summer_school_2013/NYU.ppt
Tim Maudlin
NYU
Physics & Philosophy of Time
July 25, 2013
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
In this paper, we introduce and study the notion of Zα-open sets and some properties of this class of sets are investigated. Also, we introduce the class of A *L-sets via Zα-open sets. Further, by using these sets, a new decompositions of continuous functions are presented. (2000) AMS Subject Classifications: 54D10; 54C05; 54C08.
An Overview of Separation Axioms by Nearly Open Sets in Topology.IJERA Editor
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.
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
In this paper we introduce a new class of sets known as
δ
ˆ
S –closed sets in ideal topological
spaces and we studied some of its basic properties and characterizations. This new class of sets lies
between –I–closed [19] sets and g–closed sets, and its unique feature is it forms topology and it is
independent of open sets.
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
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.
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.
Abstract:
"We study different possibilities to apply the principles of rough-paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of rough-paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelli's controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy."
René Schott
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
Impact of the Hydrographic Changing in the Open Drains Cross Sections on the ...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
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
Tim Maudlin: New Foundations for Physical GeometryArun Gupta
New Foundations for Physical Geometry
Original URL: http://www.unil.ch/webdav/site/philo/shared/summer_school_2013/NYU.ppt
Tim Maudlin
NYU
Physics & Philosophy of Time
July 25, 2013
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
In this paper, we introduce and study the notion of Zα-open sets and some properties of this class of sets are investigated. Also, we introduce the class of A *L-sets via Zα-open sets. Further, by using these sets, a new decompositions of continuous functions are presented. (2000) AMS Subject Classifications: 54D10; 54C05; 54C08.
An Overview of Separation Axioms by Nearly Open Sets in Topology.IJERA Editor
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.
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
In this paper we introduce a new class of sets known as
δ
ˆ
S –closed sets in ideal topological
spaces and we studied some of its basic properties and characterizations. This new class of sets lies
between –I–closed [19] sets and g–closed sets, and its unique feature is it forms topology and it is
independent of open sets.
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
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.
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.
Abstract:
"We study different possibilities to apply the principles of rough-paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of rough-paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelli's controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy."
René Schott
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
Impact of the Hydrographic Changing in the Open Drains Cross Sections on the ...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
A Novel Information Accountability Framework for Cloud ComputingIJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
Seasonal Variational Impact of the Physical Parameters On Mohand Rao River F...IJMER
The paper depicts the seasonal variational impact on water quality of Doon Valley . Study was
proposed to analyze the various water sample of Mohand-Rao river flowing in the Mohand Anticline in
the lower parts of Shiwalik hills in Doon Valley for physico-chemical characteristics of water quality
parameters such as pH; Temperature; Conductivity; Hardness; Alkalinity; Total Solids; Total Dissolved
Solids; Total Suspended Solids..To analyze the physical, chemical, and toxicological parameters of
Streams and rivers.
Impact of Hybrid Pass-Transistor Logic (HPTL) on Power, Delay and Area in VL...IJMER
Abstract: Power reduction is a serious concern now days. As the MOS devices are wide spread, there is
high need for circuits which consume less power, mainly for portable devices which run on batteries, like
Laptops and hand-held computers. The Pass-Transistor Logic (PTL) is a better way to implement circuits
designed for low power applications.
Analyzing the indicators walkability of cities, in order to improving urban v...IJMER
Urban design is a technique and knowledge seeking to organize and improve urban qualities
and increase the quality of citizenship life. Based on the perspectives and objectives of urban design, the
dominant intention in all urbanism activities is to reach high humanistic and social dimensions. In fact,
what give meaning to a city are the social aspects raised in recent urban activities, in addition to the
physical and visual body of it. Over the past decade the quality of the walking environment has become
a significant factor in transportation planning and design in developed countries. It is argued that the
pedestrians’ environment has been ignored in favors of automobile. The purpose of this study was to
examine the effects of walkability on property values and investment returns. Research method is
descriptive. The method of collected data is field. Also, were used questionnaire tools in order to
collecting data. On the other hand, was referred to municipality 9 region due to, studied area was
located in this urban region.
In continue, was used SWOT technique in order to analyzing questionnaire. At finally, proposed
strategies in order to improving urban space qualify.
Wavelet Based Analysis of Online Monitoring of Electrical Power by Mobile Tec...IJMER
Electrical automation is an important option for obtaining optimal solution while monitoring the electrical power consumption. While using the conventional methods the errors in continuous monitoring of power consumption is more. But the system requires not only the monitoring of the energy but also requires the analysis of the monitored energy. In this paper wavelet analysis is used for the analysis of the monitored energy/power which is monitored by GPRS technology. By using the GPRS mobile technology the energy consumption is monitored continuously and the observed data is interfaced to the computer by RS 232 port. By using MATLAB the monitored data is processed to obtain in depth analysis of the monitored power. The proposed method not only monitors the data but also provides efficient means to analyze the observed data by Wavelet Transform
Fast Data Collection with Interference and Life Time in Tree Based Wireless S...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
On Characterizations of NANO RGB-Closed Sets in NANO Topological SpacesIJMER
The purpose of this paper is to establish and derive the theorems which exhibit the
characterization of nano rgb-closed sets in nano topological space and obtain some of their interesting
properties. We also use this notion to consider new weak form of continuities with these sets.
2010 AMS classification: 54A05, 54C10.
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
A Study on Translucent Concrete Product and Its Properties by Using Optical F...IJMER
- Translucent concrete is a concrete based material with light-transferring properties,
obtained due to embedded light optical elements like Optical fibers used in concrete. Light is conducted
through the concrete from one end to the other. This results into a certain light pattern on the other
surface, depending on the fiber structure. Optical fibers transmit light so effectively that there is
virtually no loss of light conducted through the fibers. This paper deals with the modeling of such
translucent or transparent concrete blocks and panel and their usage and also the advantages it brings
in the field. The main purpose is to use sunlight as a light source to reduce the power consumption of
illumination and to use the optical fiber to sense the stress of structures and also use this concrete as an
architectural purpose of the building
Developing Cost Effective Automation for Cotton Seed DelintingIJMER
A low cost automation system for removal of lint from cottonseed is to be designed and
developed. The setup consists of stainless steel drum with stirrer in which cottonseeds having lint is mixed
with concentrated sulphuric acid. So lint will get burn. This lint free cottonseed treated with lime water to
neutralize acidic nature. After water washing this cottonseeds are used for agriculter purpose
Study & Testing Of Bio-Composite Material Based On Munja FibreIJMER
The incorporation of natural fibres such as munja fiber composites has gained
increasing applications both in many areas of Engineering and Technology. The aim of this study is to
evaluate mechanical properties such as flexural and tensile properties of reinforced epoxy composites.
This is mainly due to their applicable benefits as they are light weight and offer low cost compared to
synthetic fibre composites. Munja fibres recently have been a substitute material in many weight-critical
applications in areas such as aerospace, automotive and other high demanding industrial sectors. In
this study, natural munja fibre composites and munja/fibreglass hybrid composites were fabricated by a
combination of hand lay-up and cold-press methods. A new variety in munja fibre is the present work
the main aim of the work is to extract the neat fibre and is characterized for its flexural characteristics.
The composites are fabricated by reinforcing untreated and treated fibre and are tested for their
mechanical, properties strictly as per ASTM procedures.
Hybrid Engine (Stirling Engine + IC Engine + Electric Motor)IJMER
Hybrid engine is a combination of Stirling engine, IC engine and Electric motor. All these 3 are
connected together to a single shaft. The power source of the Stirling engine will be a Solar Panel. The aim of
this is to run the automobile using a Hybrid engine
Fabrication & Characterization of Bio Composite Materials Based On Sunnhemp F...IJMER
The present day technology demands eco-friendly developments. In this era the
composite material are playing a vital roal in different field of Engineering .The composite materials
are using as a principle materials. Nowaday the composite materials are utilizing as a important
component of engineering field .Where as the importance of the applications of composites is well
known, but thrust on the use of natural fibres in it for reinforcement has been given priority for some
times. But changing from synthetic fibres to natural fibres provides only half green-composites. A
partial green composite will be achieved if the matrix component is also eco-friendly. Keeping this in
view, a detailed literature surveyed has been carried out through various issues of the Journals
related to this field. The material systems used are sunnhemp fibres. Some epoxy and hardener has
been also added for stability and drying of the bio-composites. Various graphs and bar-charts are
super-imposed on each other for comparison among themselves and Graphs is plotted on MAT LAB
and ORIGIN 6.0 software. To determining tensile strengths, Various properties for different biocomposites
have been compared among themselves. Comparison of the behaviour of bio-composites of
this work has been also compare with other works. The bio-composites developed in this work are
likely to get applications in fall ceilings, partitions, bio-degradable packagings, automotive interiors,
sports things (e.g. rackets, nets, etc.), toys etc.
Geochemistry and Genesis of Kammatturu Iron Ores of Devagiri Formation, Sandu...IJMER
The Greenstone belts of Karnataka are enriched in BIFs in Dharwar craton, where Iron
formations are confined to the basin shelf, clearly separated from the deeper-water iron formation that
accumulated at the basin margin and flanking the marine basin. Geochemical data procured in terms of
major, trace and REE are plotted in various diagrams to interpret the genesis of BIFs. Al2O3, Fe2O3 (T),
TiO2, CaO, and SiO2 abundances and ratios show a wide variation. Ni, Co, Zr, Sc, V, Rb, Sr, U, Th,
ΣREE, La, Ce and Eu anomalies and their binary relationships indicate that wherever the terrigenous
component has increased, the concentration of elements of felsic such as Zr and Hf has gone up. Elevated
concentrations of Ni, Co and Sc are contributed by chlorite and other components characteristic of basic
volcanic debris. The data suggest that these formations were generated by chemical and clastic
sedimentary processes on a shallow shelf. During transgression, chemical precipitation took place at the
sediment-water interface, whereas at the time of regression. Iron ore formed with sedimentary structures
and textures in Kammatturu area, in a setting where the water column was oxygenated.
Experimental Investigation on Characteristic Study of the Carbon Steel C45 in...IJMER
In this paper, the mechanical characteristics of C45 medium carbon steel are investigated
under various working conditions. The main characteristic to be studied on this paper is impact toughness
of the material with different configurations and the experiment were carried out on charpy impact testing
equipment. This study reveals the ability of the material to absorb energy up to failure for various
specimen configurations under different heat treated conditions and the corresponding results were
compared with the analysis outcome
Non linear analysis of Robot Gun Support Structure using Equivalent Dynamic A...IJMER
Robot guns are being increasingly employed in automotive manufacturing to replace
risky jobs and also to increase productivity. Using a single robot for a single operation proves to be
expensive. Hence for cost optimization, multiple guns are mounted on a single robot and multiple
operations are performed. Robot Gun structure is an efficient way in which multiple welds can be done
simultaneously. However mounting several weld guns on a single structure induces a variety of
dynamic loads, especially during movement of the robot arm as it maneuvers to reach the weld
locations. The primary idea employed in this paper, is to model those dynamic loads as equivalent G
force loads in FEA. This approach will be on the conservative side, and will be saving time and
subsequently cost efficient. The approach of the paper is towards creating a standard operating
procedure when it comes to analysis of such structures, with emphasis on deploying various technical
aspects of FEA such as Non Linear Geometry, Multipoint Constraint Contact Algorithm, Multizone
meshing .
Static Analysis of Go-Kart Chassis by Analytical and Solid Works SimulationIJMER
This paper aims to do modelling, simulation and performing the static analysis of a go
kart chassis consisting of Circular beams. Modelling, simulations and analysis are performed using 3-D
modelling software i.e. Solid Works and ANSYS according to the rulebook provided by Indian Society of
New Era Engineers (ISNEE) for National Go Kart Championship (NGKC-14).The maximum deflection is
determined by performing static analysis. Computed results are then compared to analytical calculation,
where it is found that the location of maximum deflection agrees well with theoretical approximation but
varies on magnitude aspect.
In récent year various vehicle introduced in market but due to limitation in
carbon émission and BS Séries limitd speed availability vehicle in the market and causing of
environnent pollution over few year There is need to decrease dependancy on fuel vehicle.
bicycle is to be modified for optional in the future To implement new technique using change in
pedal assembly and variable speed gearbox such as planetary gear optimise speed of vehicle
with variable speed ratio.To increase the efficiency of bicycle for confortable drive and to
reduce torque appli éd on bicycle. we introduced epicyclic gear box in which transmission done
throgh Chain Drive (i.e. Sprocket )to rear wheel with help of Epicyclical gear Box to give
number of différent Speed during driving.To reduce torque requirent in the cycle with change in
the pedal mechanism
Integration of Struts & Spring & Hibernate for Enterprise ApplicationsIJMER
The proposal of this paper is to present Spring Framework which is widely used in
developing enterprise applications. Considering the current state where applications are developed using
the EJB model, Spring Framework assert that ordinary java beans(POJO) can be utilize with minimal
modifications. This modular framework can be used to develop the application faster and can reduce
complexity. This paper will highlight the design overview of Spring Framework along with its features that
have made the framework useful. The integration of multiple frameworks for an E-commerce system has
also been addressed in this paper. This paper also proposes structure for a website based on integration of
Spring, Hibernate and Struts Framework.
Microcontroller Based Automatic Sprinkler Irrigation SystemIJMER
Microcontroller based Automatic Sprinkler System is a new concept of using
intelligence power of embedded technology in the sprinkler irrigation work. Designed system replaces
the conventional manual work involved in sprinkler irrigation to automatic process. Using this system a
farmer is protected against adverse inhuman weather conditions, tedious work of changing over of
sprinkler water pipe lines & risk of accident due to high pressure in the water pipe line. Overall
sprinkler irrigation work is transformed in to a comfortableautomatic work. This system provides
flexibility & accuracy in respect of time set for the operation of a sprinkler water pipe lines. In present
work the author has designed and developed an automatic sprinkler irrigation system which is
controlled and monitored by a microcontroller interfaced with solenoid valves.
On some locally closed sets and spaces in Ideal Topological SpacesIJMER
In this paper we introduce and characterize some new generalized locally closed sets
known as
δ
ˆ
s-locally closed sets and spaces are known as
δ
ˆ
s-normal space and
δ
ˆ
s-connected space and
discussed some of their properties
Intrusion Detection and Forensics based on decision tree and Association rule...IJMER
This paper present an approach based on the combination of, two techniques using
decision tree and Association rule mining for Probe attack detection. This approach proves to be
better than the traditional approach of generating rules for fuzzy expert system by clustering methods.
Association rule mining for selecting the best attributes together and decision tree for identifying the
best parameters together to create the rules for fuzzy expert system. After that rules for fuzzy expert
system are generated using association rule mining and decision trees. Decision trees is generated for
dataset and to find the basic parameters for creating the membership functions of fuzzy inference
system. Membership functions are generated for the probe attack. Based on these rules we have
created the fuzzy inference system that is used as an input to neuro-fuzzy system. Fuzzy inference
system is loaded to neuro-fuzzy toolbox as an input and the final ANFIS structure is generated for
outcome of neuro-fuzzy approach. The experiments and evaluations of the proposed method were
done with NSL-KDD intrusion detection dataset. As the experimental results, the proposed approach
based on the combination of, two techniques using decision tree and Association rule mining
efficiently detected probe attacks. Experimental results shows better results for detecting intrusions as
compared to others existing methods
Natural Language Ambiguity and its Effect on Machine LearningIJMER
"Natural language processing" here refers to the use and ability of systems to process
sentences in a natural language such as English, rather than in a specialized artificial computer
language such as C++. The systems of real interest here are digital computers of the type we think of as
personal computers and mainframes. Of course humans can process natural languages, but for us the
question is whether digital computers can or ever will process natural languages. We have tried to
explore in depth and break down the types of ambiguities persistent throughout the natural languages
and provide an answer to the question “How it affects the machine translation process and thereby
machine learning as whole?” .
Today in era of software industry there is no perfect software framework available for
analysis and software development. Currently there are enormous number of software development
process exists which can be implemented to stabilize the process of developing a software system. But no
perfect system is recognized till yet which can help software developers for opting of best software
development process. This paper present the framework of skillful system combined with Likert scale. With
the help of Likert scale we define a rule based model and delegate some mass score to every process and
develop one tool name as MuxSet which will help the software developers to select an appropriate
development process that may enhance the probability of system success.
Material Parameter and Effect of Thermal Load on Functionally Graded CylindersIJMER
The present study investigates the creep in a thick-walled composite cylinders made
up of aluminum/aluminum alloy matrix and reinforced with silicon carbide particles. The distribution
of SiCp is assumed to be either uniform or decreasing linearly from the inner to the outer radius of
the cylinder. The creep behavior of the cylinder has been described by threshold stress based creep
law with a stress exponent of 5. The composite cylinders are subjected to internal pressure which is
applied gradually and steady state condition of stress is assumed. The creep parameters required to
be used in creep law, are extracted by conducting regression analysis on the available experimental
results. The mathematical models have been developed to describe steady state creep in the composite
cylinder by using von-Mises criterion. Regression analysis is used to obtain the creep parameters
required in the study. The basic equilibrium equation of the cylinder and other constitutive equations
have been solved to obtain creep stresses in the cylinder. The effect of varying particle size, particle
content and temperature on the stresses in the composite cylinder has been analyzed. The study
revealed that the stress distributions in the cylinder do not vary significantly for various combinations
of particle size, particle content and operating temperature except for slight variation observed for
varying particle content. Functionally Graded Materials (FGMs) emerged and led to the development
of superior heat resistant materials.
Energy Audit is the systematic process for finding out the energy conservation
opportunities in industrial processes. The project carried out studies on various energy conservation
measures application in areas like lighting, motors, compressors, transformer, ventilation system etc.
In this investigation, studied the technical aspects of the various measures along with its cost benefit
analysis.
Investigation found that major areas of energy conservation are-
1. Energy efficient lighting schemes.
2. Use of electronic ballast instead of copper ballast.
3. Use of wind ventilators for ventilation.
4. Use of VFD for compressor.
5. Transparent roofing sheets to reduce energy consumption.
So Energy Audit is the only perfect & analyzed way of meeting the Industrial Energy Conservation.
An Implementation of I2C Slave Interface using Verilog HDLIJMER
The focus of this paper is on implementation of Inter Integrated Circuit (I2C) protocol
following slave module for no data loss. In this paper, the principle and the operation of I2C bus protocol
will be introduced. It follows the I2C specification to provide device addressing, read/write operation and
an acknowledgement. The programmable nature of device provide users with the flexibility of configuring
the I2C slave device to any legal slave address to avoid the slave address collision on an I2C bus with
multiple slave devices. This paper demonstrates how I2C Master controller transmits and receives data to
and from the Slave with proper synchronization.
The module is designed in Verilog and simulated in ModelSim. The design is also synthesized in Xilinx
XST 14.1. This module acts as a slave for the microprocessor which can be customized for no data loss.
Discrete Model of Two Predators competing for One PreyIJMER
This paper investigates the dynamical behavior of a discrete model of one prey two
predator systems. The equilibrium points and their stability are analyzed. Time series plots are obtained
for different sets of parameter values. Also bifurcation diagrams are plotted to show dynamical behavior
of the system in selected range of growth parameter
Discrete Model of Two Predators competing for One Prey
Ac2640014009
1. International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4001-4009 ISSN: 2249-6645
On rg-Separation Axioms
S. Balasubramanian1 C. Sandhya2 and M.D.S. Saikumar3
1
Department of Mathematics, Govt. Arts College (A), Karur – 639 005, Tamilnadu
2
Department of Mathematics, C.S.R. Sarma College, Ongole – 523 001, Andhraparadesh
3
Department of Mathematics, Krishnaveni Degree College, Narasaraopet – 522 601, Andhraparadesh
Abstract: In this paper we define almost rg-normality and mild rg-normality, continue the study of further properties of rg-
normality. We show that these three axioms are regular open hereditary. Also define the class of almost rg-irresolute
mappings and show that rg-normality is invariant under almost rg-irresolute M-rg-open continuous surjection.
AMS Subject Classification: 54D15, 54D10.
Key words and Phrases: rg-open, almost normal, midly normal, M-rg-closed, M-rg-open, rc-continuous.
I. Introduction:
In 1967, A. Wilansky has introduced the concept of US spaces. In 1968, C.E. Aull studied some separation axioms
between the T1 and T2 spaces, namely, S1 and S2. Next, in 1982, S.P. Arya et al have introduced and studied the concept of
semi-US spaces and also they made study of s-convergence, sequentially semi-closed sets, sequentially s-compact notions.
G.B. Navlagi studied P-Normal Almost-P-Normal, Mildly-P-Normal and Pre-US spaces. Recently S. Balasubramanian and
P.Aruna Swathi Vyjayanthi studied v-Normal Almost- v-Normal, Mildly-v-Normal and v-US spaces. Inspired with these we
introduce rg-Normal Almost- rg-Normal, Mildly- rg-Normal, rg-US, rg-S1 and rg-S2. Also we examine rg-convergence,
sequentially rg-compact, sequentially rg-continuous maps, and sequentially sub rg-continuous maps in the context of these
new concepts. All notions and symbols which are not defined in this paper may be found in the appropriate references.
Throughout the paper X and Y denote Topological spaces on which no separation axioms are assumed explicitly stated.
II. Preliminaries:
Definition 2.1: AX is called g-closed[resp: rg-closed] if clAU[resp: scl(A) U] whenever A U and U is open[resp:
semi-open] in X.
Definition 2.2: A space X is said to be
(i) T1(T2) if for x y in X, there exist (disjoint) open sets U; V in X such that xU and yV.
(ii) weakly Hausdorff if each point of X is the intersection of regular closed sets of X.
(iii) Normal [resp: mildly normal] if for any pair of disjoint [resp: regular-closed] closed sets F1 and F2 , there exist disjoint
open sets U and V such that F1 U and F2 V.
(iv) almost normal if for each closed set A and each regular closed set B such that AB = , there exist disjoint open sets U
and V such that AU and BV.
(v) weakly regular if for each pair consisting of a regular closed set A and a point x such that A {x} = , there exist
disjoint open sets U and V such that x U and AV.
(vi) A subset A of a space X is S-closed relative to X if every cover of A by semi-open sets of X has a finite subfamily
whose closures cover A.
(vii) R0 if for any point x and a closed set F with xF in X, there exists a open set G containing F but not x.
(viii) R1 iff for x, y X with cl{x} cl{y}, there exist disjoint open sets U and V such that cl{x} U, cl{y}V.
(ix) US-space if every convergent sequence has exactly one limit point to which it converges. (x) pre-US space if every pre-
convergent sequence has exactly one limit point to which it converges.
(xi) pre-S1 if it is pre-US and every sequence pre-converges with subsequence of pre-side points.
(xii) pre-S2 if it is pre-US and every sequence in X pre-converges which has no pre-side point.
(xiii) is weakly countable compact if every infinite subset of X has a limit point in X.
(xiv) Baire space if for any countable collection of closed sets with empty interior in X, their union also has empty interior in
Definition 2.3: Let A X. Then a point x is said to be a
(i) limit point of A if each open set containing x contains some point y of A such that x y.
(ii) T0–limit point of A if each open set containing x contains some point y of A such that cl{x} cl{y}, or equivalently, such
that they are topologically distinct.
(iii) pre-T0–limit point of A if each open set containing x contains some point y of A such that pcl{x} pcl{y}, or
equivalently, such that they are topologically distinct.
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Note 1: Recall that two points are topologically distinguishable or distinct if there exists an open set containing one of the
points but not the other; equivalently if they have disjoint closures. In fact, the T 0–axiom is precisely to ensure that any two
distinct points are topologically distinct.
Example 1: Let X = {a, b, c, d} and τ = {{a}, {b, c}, {a, b, c}, X, }. Then b and c are the limit points but not the T 0–limit
points of the set {b, c}. Further d is a T0–limit point of {b, c}.
Example 2: Let X = (0, 1) and τ = {, X, and Un = (0, 1–1⁄n), n = 2, 3, 4,. . . }. Then every point of X is a limit point of X.
Every point of XU2 is a T0–limit point of X, but no point of U2 is a T0–limit point of X.
Definition 2.4: A set A together with all its T0–limit points will be denoted by T0–clA.
Note 2: i. Every T0–limit point of a set A is a limit point of the set but converse is not true.
ii. In T0–space both are same.
Note 3: R0–axiom is weaker than T1–axiom. It is independent of the T0–axiom. However T1 = R0+T0
Note 4: Every countable compact space is weakly countable compact but converse is not true in general. However, a T 1–
space is weakly countable compact iff it is countable compact.
Definition 3.01: In X, a point x is said to be a rg-T0–limit point of A if each rg-open set containing x contains some point y
of A such that rgcl{x} rgcl{y}, or equivalently; such that they are topologically distinct with respect to rg-open sets.
III. Example
Let X = {a, b, c} and = {, b, a, b, b, c, X. For A = {a, b}, a is rg-T0–limit point.
Definition 3.02: A set A together with all its rg-T0–limit points is denoted by T0-rgcl (A)
Lemma 3.01: If x is a rg-T0–limit point of a set A then x is rg-limit point of A.
Lemma 3.02: If X is rgT0 [resp: rT0–]–space then every rg-T0–limit point and every rg-limit point are equivalent.
Theorem 3.03: For x ≠ y X,
(i) X is a rg-T0–limit point of {y} iff xrgcl{y} and yrgcl{x}.
(ii) X is not a rg-T0–limit point of {y} iff either xrgcl {y} or rgcl{x} = rgcl{y}.
(iii) X is not a rg-T0–limit point of {y} iff either xrgcl{y} or yrgcl{x}.
Corollary 3.04:
(i) If x is a rg-T0–limit point of {y}, then y cannot be a rg-limit point of {x}.
(ii) If rgcl{x} = rgcl{y}, then neither x is a rg-T0–limit point of {y} nor y is a rg-T0–limit point of {x}.
(iii) If a singleton set A has no rg-T0–limit point in X, then rgclA = rgcl{x} for all x rgcl{A}.
Lemma 3.05: In X, if x is a rg-limit point of a set A, then in each of the following cases x becomes rg-T0–limit point of A ({x}
≠ A).
(i) rgcl{x} rgcl{y} for yA, x y.
(ii) rgcl{x} = {x}
(iii) X is a rg-T0–space.
(iv) A{x} is rg-open
IV. rg-T0 AND rg-Ri AXIOMS, i = 0,1:
In view of Lemma 3.5(iii), rg-T0–axiom implies the equivalence of the concept of limit point with that of rg-T0–
limit point of the set. But for the converse, if x rgcl{y} then rgcl{x} ≠ rgcl{y} in general, but if x is a rg-T0–limit point of
{y}, then rgcl{x} = rgcl{y}
Lemma 4.01: In X, a limit point x of {y} is a rg-T0–limit point of {y} iff rgcl{x} ≠ rgcl{y}.
This lemma leads to characterize the equivalence of rg-T0–limit point and rg-limit point of a set as rg-T0–axiom.
Theorem 4.02: The following conditions are equivalent:
(i) X is a rg-T0 space
(ii) Every rg-limit point of a set A is a rg-T0–limit point of A
(iii) Every r-limit point of a singleton set {x} is a rg-T0–limit point of {x}
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(iv) For any x, y in X, x ≠ y if x rgcl{y}, then x is a rg-T0–limit point of {y}
Note 5: In a rg-T0–space X, if every point of X is a r-limit point, then every point is rg-T0–limit point. But if each point is a
rg-T0–limit point of X it is not necessarily a rg-T0–space
Theorem 4.03: The following conditions are equivalent:
(i) X is a rg-R0 space
(ii) For any x, y in X, if x rgcl{y}, then x is not a rg-T0–limit point of {y}
(iii) A point rg-closure set has no rg-T0–limit point in X
(iv) A singleton set has no rg-T0–limit point in X.
Theorem 4.04: In a rg-R0 space X, a point x is rg-T0–limit point of A iff every rg-open set containing x contains infinitely
many points of A with each of which x is topologically distinct
Theorem 4.05: X is rg-R0 space iff a set A of the form A = rgcl{xi i =1 to n} a finite union of point closure sets has no rg-T0–
limit point.
Corollary 4.06: The following conditions are equivalent:
(i) X is a rR0 space
(ii) For any x, y in X, if x rgcl{y}, then x is not a rg-T0–limit point of {y}
(iii) A point rg-closure set has no rg-T0–limit point in X
(iv) A singleton set has no rg-T0–limit point in X.
Corollary 4.07: In an rR0–space X,
(i) If a point x is rg-T0–[resp:rT0–] limit point of a set then every rg-open set containing x contains infinitely many points of
A with each of which x is topologically distinct.
(ii) If A = rgcl{xi, i =1 to n} a finite union of point closure sets has no rg-T0–limit point.
(iii) If X = rgcl{xi, i =1 to n} then X has no rg-T0–limit point.
Various characteristic properties of rg-T0–limit points studied so far is enlisted in the following theorem.
Theorem 4.08: In a rg-R0–space, we have the following:
(i) A singleton set has no rg-T0–limit point in X.
(ii) A finite set has no rg-T0–limit point in X.
(iii) A point rg-closure has no set rg-T0–limit point in X
(iv) A finite union point rg-closure sets have no set rg-T0–limit point in X.
(v) For x, y X, xT0– rgcl{y} iff x = y.
(vi) x ≠ y X, iff neither x is rg-T0–limit point of {y}nor y is rg-T0–limit point of {x}
(vii) For any x, y X, x ≠ y iff T0– rgcl{x} T0– rgcl{y} = .
(viii) Any point xX is a rg-T0–limit point of a set A in X iff every rg-open set containing x contains infinitely many
points of A with each which x is topologically distinct.
Theorem 4.09: X is rg-R1 iff for any rg-open set U in X and points x, y such that x XU, yU, there exists a rg-open set V
in X such that yVU, xV.
Lemma 4.10: In rg-R1 space X, if x is a rg-T0–limit point of X, then for any non empty rg-open set U, there exists a non
empty rg-open set V such that VU, x rgcl(V).
Lemma 4.11: In a rg- regular space X, if x is a rg-T0–limit point of X, then for any non empty rg-open set U, there exists a
non empty rg-open set V such that rgcl(V)U, x rgcl(V).
Corollary 4.12: In a regular space X, If x is a rg-T0–[resp: T0–]limit point of X, then for any URGO(X), there exists a
non empty rg-open set V such that rgcl(V)U, x rgcl(V).
Theorem 4.13: If X is a rg-compact rg-R1-space, then X is a Baire Space.
Proof: Routine
Corollary 4.14: If X is a compact rg-R1-space, then X is a Baire Space.
Corollary 4.15: Let X be a rg-compact rg-R1-space. If {An} is a countable collection of rg-closed sets in X, each An having
non-empty rg-interior in X, then there is a point of X which is not in any of the An.
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Corollary 4.16: Let X be a rg-compact R1-space. If {An} is a countable collection of rg-closed sets in X, each An having non-
empty rg- interior in X, then there is a point of X which is not in any of the A n.
Theorem 4.17: Let X be a non empty compact rg-R1-space. If every point of X is a rg-T0–limit point of X then X is
uncountable.
Proof: Since X is non empty and every point is a rg-T0-limit point of X, X must be infinite. If X is countable, we construct a
sequence of rg-open sets {Vn} in X as follows:
Let X = V1, then for x1 is a rg-T0-limit point of X, we can choose a non empty rg-open set V2 in X such that V2 V1 and x1
rgclV2. Next for x2 and non empty rg-open set V2, we can choose a non empty rg-open set V3 in X such that V3 V2 and x2
rgclV3. Continuing this process for each xn and a non empty rg-open set Vn, we can choose a non empty rg-open set Vn+1 in
X such that Vn+1 Vn and xn rgclVn+1.
Now consider the nested sequence of rg-closed sets rgclV1 rgclV2 rgclV3 ……… rgclVn . . . Since X is
rg-compact and {rgclVn} the sequence of rg-closed sets satisfies finite intersection property. By Cantors intersection
theorem, there exists an x in X such that x rgclVn. Further xX and xV1, which is not equal to any of the points of X.
Hence X is uncountable.
Corollary 4.18: Let X be a non empty rg-compact rg-R1-space. If every point of X is a rg-T0–limit point of X then X is
uncountable
V. rg–T0-IDENTIFICATION SPACES AND rg–SEPARATION AXIOMS
Definition 5.01: Let be the equivalence relation on X defined by xy iff rgcl{x} = rgcl{y}
Problem 5.02: show that xy iff rgcl{x} = rgcl{y} is an equivalence relation
Definition 5.03: (X0, Q(X0)) is called the rg-T0–identification space of (X, ), where X0 is the set of equivalence classes of
and Q(X0) is the decomposition topology on X0.
Let PX: (X, ) (X0, Q(X0)) denote the natural map
Lemma 5.04: If xX and A X, then x rgclA iff every rg-open set containing x intersects A.
Theorem 5.05: The natural map PX:(X,) (X0, Q(X0)) is closed, open and PX –1(PX(O)) = O for all OPO(X,) and (X0,
Q(X0)) is rg-T0
Proof: Let OPO(X, ) and C PX(O). Then there exists xO such that PX(x) = C. If yC, then rgcl{y} = rgcl{x}, which
implies yO. Since PO(X,), then PX –1(PX(U)) = U for all U, which implies PX is closed and open.
Let G, HX0 such that G H; let xG and yH. Then rgcl{x} rgcl{y}, which implies xrgcl{y} or yrgcl{x}, say
xrgcl{y}. Since PX is continuous and open, then GA = PX{Xrgcl{y}}PO(X0, Q(X0)) and HA
Theorem 5.06: The following are equivalent:
(i) X is rgR0 (ii) X0 = {rgcl{x}: xX} and (iii) (X0, Q(X0)) is rgT1
Proof: (i) (ii) Let xCX0. If yC, then yrgcl{y} = rgcl{x}, which implies Crgcl{x}. If yrgcl{x}, then xrgcl{y},
since, otherwise, xXrgcl{y}PO(X,) which implies rgcl{x}Xrgcl{y}, which is a contradiction. Thus, if yrgcl{x},
then xrgcl{y}, which implies rgcl{y} = rgcl{x} and yC. Hence X0 = {rgcl{x}: xX}
(ii)(iii) Let A BX0. Then there exists x, yX such that A = rgcl{x}; B = rgcl{y}, and rgcl{x}rgcl{y} = . Then AC
= PX (Xrgcl{y})PO(X0, Q(X0)) and BC. Thus (X0, Q(X0)) is rg-T1
(iii) (i) Let xURGO(X). Let yU and Cx, Cy X0 containing x and y respectively. Then x rgcl{y}, implies Cx Cy
and there exists rg-open set A such that CxA and CyA. Since PX is continuous and open, then yB = PX–1(A) xRGO(X)
and xB, which implies yrgcl{x}. Thus rgcl{x} U. This is true for all rgcl{x} implies rgcl{x} U. Hence X is rg-R0
Theorem 5.07: (X, ) is rg-R1 iff (X0, Q(X0)) is rg-T2
The proof is straight forward using theorems 5.05 and 5.06 and is omitted
Theorem 5.08: X is rg-Ti ; i = 0,1,2. iff there exists a rg-continuous, almost–open, 1–1 function from X into a rg-Ti space ; i
= 0,1,2. respectively.
Theorem 5.09: If is rg-continuous, rg-open, and x, yX such that rgcl{x} = rgcl{y}, then rgcl{(x)} = rgcl{(y)}.
Theorem 5.10: The following are equivalent
(i) X is rg-T0
(ii) Elements of X0 are singleton sets and
(iii)There exists a rg-continuous, rg-open, 1–1 function:X Y, where Y is rg-T0
Proof: (i) is equivalent to (ii) and (i) (iii) are straight forward and is omitted.
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(iii) (i) Let x, yX such that (x) (y), which implies rgcl{(x)} rgcl{(y)}. Then by theorem 5.09, rgcl{x}
rgcl{y}. Hence (X, ) is rg-T0
Corollary 5.11: X is rg-Ti ; i = 1,2 iff X is rg-Ti –- 1 ; i = 1,2, respectively, and there exists a rg-continuous , rg-open, 1–1
function :X into a rg-T0 space.
Definition 5.04: is point–rg-closure 1–1 iff for x, yX such that rgcl{x} rgcl{y}, rgcl{(x)} rgcl{(y)}.
Theorem 5.12: (i)If :X Y is point– rg-closure 1–1 and (X, ) is rg-T0 , then is 1–1
(ii)If:X Y, where X and Y are rg-T0 then is point– rg-closure 1–1 iff is 1–1
The following result can be obtained by combining results for rg-T0– identification spaces, rg-induced functions and rg-Ti
spaces; i = 1,2.
Theorem 5.13: X is rg-Ri ; i = 0,1 iff there exists a rg-continuous , almost–open point– rg-closure 1–1 function : (X, )
into a rg-Ri space; i = 0,1 respectively.
VI. rg-Normal; Almost rg-normal and Mildly rg-normal spaces
Definition 6.1: A space X is said to be rg-normal if for any pair of disjoint closed sets F1 and F2 , there exist disjoint rg-open
sets U and V such that F1 U and F2 V.
Example 4: Let X = {a, b, c} and τ = {φ, {a}, {b, c}, X }. Then X is rg-normal.
Example 5: Let X = {a, b, c, d} and τ = {φ, {b, d}, {a, b, d}, {b, c, d}, X}. Then X is rg-normal and is not
normal.
Example 6: Let X = a, b, c, d with = {, {a}, {b}, {d}, {a, b}, {a, d}, {b, d}, {a, b, c}, {a, b, d}, X} is rg-normal,
normal and almost normal.
We have the following characterization of rg-normality.
Theorem 6.1: For a space X the following are equivalent:
(i) X is rg-normal.
(ii) For every pair of open sets U and V whose union is X, there exist rg-closed sets A and B such that AU, B V and AB
= X.
(iii) For every closed set F and every open set G containing F, there exists a rg-open set U such that FUrgcl(U)G.
Proof: (i)(ii): Let U and V be a pair of open sets in a rg-normal space X such that X = UV. Then X–U, X–V are disjoint
closed sets. Since X is rg-normal there exist disjoint rg-open sets U1 and V1 such that X–UU1 and X-VV1. Let A = X–U1, B
= X–V1. Then A and B are rg-closed sets such that AU, BV and AB = X.
(ii) (iii): Let F be a closed set and G be an open set containing F. Then X–F and G are open sets whose union is X. Then
by (b), there exist rg-closed sets W1 and W2 such that W1 X–F and W2 G and W1W2 = X. Then F X–W1, X–G X–
W2 and (X–W1)(X–W2) = . Let U = X–W1 and V= X–W2. Then U and V are disjoint rg-open sets such that FUX–VG.
As X–V is rg-closed set, we have rgcl(U) X–V and FUrgcl(U)G.
(iii) (i): Let F1 and F2 be any two disjoint closed sets of X. Put G = X–F2, then F1G = . F1G where G is an open set.
Then by (c), there exists a rg-open set U of X such that F1 U rgcl(U) G. It follows that F2 X–rgcl(U) = V, say, then
V is rg-open and UV = . Hence F1 and F2 are separated by rg-open sets U and V. Therefore X is rg-normal.
Theorem 6.2: A regular open subspace of a rg-normal space is rg-normal.
Definition 6.2: A function f:XY is said to be almost–rg-irresolute if for each x in X and each rg-neighborhood V of f(x),
rgcl(f –1(V)) is a rg-neighborhood of x.
Clearly every rg-irresolute map is almost rg-irresolute.
The Proof of the following lemma is straightforward and hence omitted.
Lemma 6.1: f is almost rg-irresolute iff f-1(V) rg-int(rgcl(f-1(V)))) for every VRGO(Y).
Lemma 6.2: f is almost rg-irresolute iff f(rgcl(U)) rgcl(f(U)) for every URGO(X).
Proof: Let URGO(X). If yrgcl(f(U)). Then there exists V RGO(y) such that Vf(U) = . Hence f -1(V)U= . Since
URGO(X), we have rg-int(rgcl(f-1(V)))rgcl(U) = . By lemma 6.1, f -1(V) rgcl(U) = and hence Vf(rgcl(U)) = .
This implies that yf(rgcl(U)).
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Conversely, if VRGO(Y), then W = X- rgcl(f-1(V))) RGO(X). By hypothesis, f(rgcl(W)) rgcl (f(W))) and hence X- rg-
int(rgcl(f-1(V))) = rgcl(W)f-1(rgcl(f(W)))f(rgcl[f(X-f-1(V))]) f –1[rgcl(Y-V)] = f -1(Y-V) = X-f-1(V). Therefore f-1(V) rg-
int(rgcl(f-1(V))). By lemma 6.1, f is almost rg-irresolute.
Theorem 6.3: If f is M-rg-open continuous almost rg-irresolute, X is rg-normal, then Y is rg-normal.
Proof: Let A be a closed subset of Y and B be an open set containing A. Then by continuity of f, f-1(A) is closed and f-1(B) is
an open set of X such that f-1 (A) f-1(B). As X is rg-normal, there exists a rg-open set U in X such that f-1(A) U
rgcl(U) f-1(B). Then f(f-1(A)) f(U) f(rgcl(U)) f(f-1(B)). Since f is M-rg-open almost rg-irresolute surjection, we obtain
A f(U) rgcl(f(U)) B. Then again by Theorem 6.1 the space Y is rg-normal.
Lemma 6.3: A mapping f is M-rg-closed iff for each subset B in Y and for each rg-open set U in X containing f-1(B), there
exists a rg-open set V containing B such that f-1(V)U.
Theorem 6.4: If f is M-rg-closed continuous, X is rg-normal space, then Y is rg-normal.
Proof of the theorem is routine and hence omitted.
Theorem 6.5: If f is an M-rg-closed map from a weakly Hausdorff rg-normal space X onto a space Y such that f-1(y) is S-
closed relative to X for each yY, then Y is rg-T2.
Proof: Let y1 y2Y. Since X is weakly Hausdorff, f -1(y1) and f -1(y2) are disjoint closed subsets of X by lemma 2.2 [12.].
As X is rg-normal, there exist disjoint Vi RGO(X, f -1(yi)) for i = 1, 2. Since f is M-rg-closed, there exist disjoint
UiRGO(Y, yi) and f -1(Ui) Vi for i = 1, 2. Hence Y is rg-T2.
Theorem 6.6: For a space X we have the following:
(a) If X is normal then for any disjoint closed sets A and B, there exist disjoint rg-open sets U, V such that AU and BV;
(b) If X is normal then for any closed set A and any open set V containing A, there exists an rg-open set U of X such that
AUrgcl(U) V.
Definition 6.2: X is said to be almost rg-normal if for each closed set A and each regular closed set B with AB = , there
exist disjoint U; VRGO(X) such that AU and BV.
Clearly, every rg-normal space is almost rg-normal, but not conversely in general.
Example 7: Let X = {a, b, c} and τ = {φ, {a}, {a, b}, {a, c}, X}. Then X is almost rg-normal and rg-
normal.
Theorem 6.7: For a space X the following statements are equivalent:
(i) X is almost rg-normal
(ii) For every pair of sets U and V, one of which is open and the other is regular open whose union is X, there exist rg-closed
sets G and H such that GU, HV and GH = X.
(iii) For every closed set A and every regular open set B containing A, there is a rg-open set V such that AVrgcl(V)B.
Proof: (i)(ii) Let U and VRO(X) such that UV = X. Then (X-U) is closed set and (X-V) is regular closed set with
(X-U)(X-V) = . By almost rg-normality of X, there exist disjoint rg-open sets U1 and V1 such that X-U U1 and X-V
V1. Let G = X- U1 and H = X-V1. Then G and H are rg-closed sets such that GU, HV and GH = X.
(ii) (iii) and (iii) (i) are obvious.
One can prove that almost rg-normality is also regular open hereditary.
Almost rg-normality does not imply almost rg-regularity in general. However, we observe that every almost rg-normal rg-R0
space is almost rg-regular.
Theorem 6.8: Every almost regular, rg-compact space X is almost rg-normal.
Recall that a function f : X Y is called rc-continuous if inverse image of regular closed set is regular closed.
Theorem 6.9: If f is continuous M-rg-open rc-continuous and almost rg-irresolute surjection from an almost rg-normal space
X onto a space Y, then Y is almost rg-normal.
Definition 6.3: X is said to be mildly rg-normal if for every pair of disjoint regular closed sets F1 and F2 of X, there exist
disjoint rg-open sets U and V such that F1 U and F2 V.
Example 8: Let X = a, b, c, d with = {, {a}, {b}, {d}, {a, b}, {a, d}, {b, d}, {a, b, c}, {a, b, d}, X} is Mildly rg-normal.
Theorem 6.10: For a space X the following are equivalent.
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(i) X is mildly rg-normal.
(ii) For every pair of regular open sets U and V whose union is X, there exist rg-closed sets G and H such that G U, H
V and GH = X.
(iii) For any regular closed set A and every regular open set B containing A, there exists a rg-open set U such that
AUrgcl(U)B.
(iv) For every pair of disjoint regular closed sets, there exist rg-open sets U and V such that AU, BV and rgcl(U)
rgcl(V) = .
Proof: This theorem may be proved by using the arguments similar to those of Theorem 6.7.
Also, we observe that mild rg-normality is regular open hereditary.
Definition 6.4: A space X is weakly rg-regular if for each point x and a regular open set U containing {x}, there is a rg-open
set V such that xV clV U.
Example 9: Let X = {a, b, c} and = {, b,a, b,b, c, X. Then X is weakly rg-regular.
Example 10: Let X = {a, b, c} and = {, a,b,a, b, X. Then X is not weakly rg-regular.
Theorem 6.11: If f : X Y is an M-rg-open rc-continuous and almost rg-irresolute function from a mildly rg-normal space
X onto a space Y, then Y is mildly rg-normal.
Proof: Let A be a regular closed set and B be a regular open set containing A. Then by rc-continuity of f, f –1(A) is a
regular closed set contained in the regular open set f-1(B). Since X is mildly rg-normal, there exists a rg-open set V such that
f-1(A) V rgcl(V) f –1(B) by Theorem 6.10. As f is M-rg-open and almost rg-irresolute surjection, f(V)RGO(Y) and
A f(V) rgcl(f(V)) B. Hence Y is mildly rg-normal.
Theorem 6.12: If f:XY is rc-continuous, M-rg-closed map and X is mildly rg-normal space, then Y is mildly rg-normal.
VII. rg-US spaces:
Definition 7.1: A point y is said to be a
(i) rg-cluster point of sequence <xn> iff <xn> is frequently in every rg-open set containing x. The set of all rg-cluster points
of <xn> will be denoted by rg-cl(xn).
(ii) rg-side point of a sequence <xn> if y is a rg-cluster point of <xn> but no subsequence of <xn> rg-converges to y.
Definition 7.2:A sequence <xn> is said to be rg-converges to a point x of X, written as <xn> rg x if <xn> is eventually in
every rg-open set containing x.
Clearly, if a sequence <xn> r-converges to a point x of X, then <xn> rg-converges to x.
Definition 7.3: A subset F is said to be
(i) sequentially rg-closed if every sequence in F rg-converges to a point in F.
(ii) sequentially rg-compact if every sequence in F has a subsequence which rg-converges to a point in F.
Definition 7.4: X is said to be
(i) rg-US if every sequence <xn> in X rg-converges to a unique point.
(ii) rg-S1 if it is rg-US and every sequence <xn> rg-converges with subsequence of <xn> rg-side points.
(iii) rg-S2 if it is rg-US and every sequence <xn> in X rg-converges which has no rg-side point.
Definition 7.5: A function f is said to be sequentially rg-continuous at xX if f(xn) rg f(x) whenever <xn>rg x. If f is
sequentially rg-continuous at all xX, then f is said to be sequentially rg-continuous.
Theorem 7.1: We have the following:
(i) Every rg-T2 space is rg-US.
(ii) Every rg-US space is rg-T1.
(iii) X is rg-US iff the diagonal set is a sequentially rg-closed subset of X x X.
(iv) X is rg-T2 iff it is both rg-R1 and rg-US.
(v) Every regular open subset of a rg-US space is rg-US.
(vi) Product of arbitrary family of rg-US spaces is rg-US.
(vii) Every rg-S2 space is rg-S1 and every rg-S1 space is rg-US.
Theorem 7.2: In a rg-US space every sequentially rg-compact set is sequentially rg-closed.
Proof: Let X be rg-US space. Let Y be a sequentially rg-compact subset of X. Let <xn> be a sequence in Y. Suppose that
<xn> rg-converges to a point in X-Y. Let <xnp> be subsequence of <xn> that rg-converges to a point y Y since Y is
sequentially rg-compact. Also, let a subsequence <xnp> of <xn> rg-converge to x X-Y. Since <xnp> is a sequence in the
rg-US space X, x = y. Thus, Y is sequentially rg-closed set.
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Theorem 7.3: If f and g are sequentially rg-continuous and Y is rg-US, then the set A = {x | f(x) = g(x)} is sequentially rg-
closed.
Proof: Let Y be rg-US. If there is a sequence <xn> in A rg-converging to x X. Since f and g are sequentially rg-
continuous, f(xn) rg f(x) and g(xn) rg g(x). Hence f(x) = g(x) and xA. Therefore, A is sequentially rg-closed.
VIII. Sequentially sub-rg-continuity:
Definition 8.1: A function f is said to be
(i) sequentially nearly rg-continuous if for each point xX and each sequence <xn> rg x in X, there exists a subsequence
<xnk> of <xn> such that <f(xnk)> rg f(x).
(ii) sequentially sub-rg-continuous if for each point xX and each sequence <xn> rg x in X, there exists a subsequence
<xnk> of <xn> and a point yY such that <f(xnk)> rg y.
(iii) sequentially rg-compact preserving if f(K) is sequentially rg-compact in Y for every sequentially rg-compact set K of X.
Lemma 8.1: Every function f is sequentially sub-rg-continuous if Y is a sequentially rg-compact.
Proof: Let <xn> rg x in X. Since Y is sequentially rg-compact, there exists a subsequence {f(xnk)} of {f(xn)} rg-converging
to a point yY. Hence f is sequentially sub-rg-continuous.
Theorem 8.1: Every sequentially nearly rg-continuous function is sequentially rg-compact preserving.
Proof: Assume f is sequentially nearly rg-continuous and K any sequentially rg-compact subset of X. Let <yn> be any
sequence in f (K). Then for each positive integer n, there exists a point xn K such that f(xn) = yn. Since <xn> is a sequence
in the sequentially rg-compact set K, there exists a subsequence <xnk> of <xn> rg-converging to a point x K. By
hypothesis, f is sequentially nearly rg-continuous and hence there exists a subsequence <xj> of <xnk> such that f(xj) rg f(x).
Thus, there exists a subsequence <yj> of <yn> rg-converging to f(x)f(K). This shows that f(K) is sequentially rg-compact
set in Y.
Theorem 8.2: Every sequentially s-continuous function is sequentially rg-continuous.
Proof: Let f be a sequentially s-continuous and <xn> s xX. Then <xn> s x. Since f is sequentially s-continuous, f(xn)s
f(x). But we know that <xn>s x implies <xn> rg x and hence f(xn) rg f(x) implies f is sequentially rg-continuous.
Theorem 8.3: Every sequentially rg-compact preserving function is sequentially sub-rg-continuous.
Proof: Suppose f is a sequentially rg-compact preserving function. Let x be any point of X and <xn> any sequence in X rg-
converging to x. We shall denote the set {xn | n= 1,2,3, …} by A and K = A {x}. Then K is sequentially rg-compact since
(xn) rg x. By hypothesis, f is sequentially rg-compact preserving and hence f(K) is a sequentially rg-compact set of Y. Since
{f(xn)} is a sequence in f(K), there exists a subsequence {f(xnk)} of {f(xn)} rg-converging to a point yf(K). This implies that
f is sequentially sub-rg-continuous.
Theorem 8.4: A function f: X Y is sequentially rg-compact preserving iff f/K: K f(K) is sequentially sub-rg-continuous
for each sequentially rg-compact subset K of X.
Proof: Suppose f is a sequentially rg-compact preserving function. Then f(K) is sequentially rg-compact set in Y for each
sequentially rg-compact set K of X. Therefore, by Lemma 8.1 above, f/K: K f(K) is sequentially rg-continuous function.
Conversely, let K be any sequentially rg-compact set of X. Let <yn> be any sequence in f(K). Then for each positive integer
n, there exists a point xnK such that f(xn) = yn. Since <xn> is a sequence in the sequentially rg-compact set K, there exists a
subsequence <xnk> of <xn> rg-converging to a point x K. By hypothesis, f /K: K f(K) is sequentially sub-rg-continuous
and hence there exists a subsequence <ynk> of <yn> rg-converging to a point y f(K).This implies that f(K) is sequentially
rg-compact set in Y. Thus, f is sequentially rg-compact preserving function.
The following corollary gives a sufficient condition for a sequentially sub-rg-continuous function to be sequentially rg-
compact preserving.
Corollary 8.1: If f is sequentially sub-rg-continuous and f(K) is sequentially rg-closed set in Y for each sequentially rg-
compact set K of X, then f is sequentially rg-compact preserving function.
IX. Acknowledgments:
The authors would like to thank the referees for their critical comments and suggestions for the development of this paper.
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