This document introduces the concept of nano continuous functions between nano topological spaces. It defines nano continuous functions as those whose inverse image of every nano-open set is nano-open. Several characterizations of nano continuous functions are provided in terms of nano closed sets, nano closure, and nano interior. Nano continuous functions are shown to preserve nano closed sets under inverse image and nano closure under inverse image composition. The concept of a basis for a nano topology is used to provide another characterization of nano continuous functions in terms of the inverse image of basis elements.
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.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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 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.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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
Minimal M-gs Open and Maximal M-gs Closed Sets In Interior Minimal Spaceinventionjournals
The main objective of this paper is to study the notions of Minimal M-GS Closed set, Maximal M-GS
Open set, Minimal M-GS Open set and Maximal M-GS Closed set and their basic properties in Interior Minimal
Space.
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.
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Field Induced Josephson Junction (FIJJ) is defined as the physical system made by placement of ferromagnetic strip directly or indirectly [insulator layer in-between] on the top of superconducting strip [3, 4, 7]. The analysis conducted in extended Ginzburg-Landau, Bogoliubov-de Gennes and RCSJ [11] models essentially points that the system is in most case a weak-link Josephson junction [2] and sometimes has features of tunneling Josephson junction [1]. Generalization of Field Induced Josephson junctions leads to the case of network of robust coupled field induced Josephson junctions [4] that interact in inductive way. Also the scheme of superconducting Random Access Memory (RAM) for Rapid Single Flux [8, 9] quantum (RSFQ) computer is drawn [6, 10] using the concept of tunneling Josephson junction [1] and Field Induced Josephson junction [3, 4].
The given presentation is also available by YouTube (https://www.youtube.com/watch?v=uIqXqiwDsSM).
Literature
[1]. B.D.Josephson, Possible new effects in superconductive tunnelling, PL, Vol.1, No. 251, 1962
[2]. K.Likharev, Josephson junctions Superconducting weak links, RMP, Vol. 51, No. 101, 1979
[3]. K.Pomorski and P.Prokopow, Possible existence of field induced Josephson junctions, PSS B, Vol.249, No.9, 2012
[4]. K.Pomorski, PhD thesis: Physical description of unconventional Josephson junction, Jagiellonian University, 2015
[4]. K.Pomorski, H.Akaike, A.Fujimaki, Towards robust coupled field induced Josephson junctions, arxiv:1607.05013, 2016
[6]. K.Pomorski, H.Akaike, A.Fujimaki, Relaxation method in description of RAM memory cell in RSFQ computer, Procedings of Applied Conference 2016 (in progress)
[7]. J.Gelhausen and M.Eschrig, Theory of a weak-link superconductor-ferromagnet Josephson structure, PRB, Vol.94, 2016
[8]. K.K. Likharev, Rapid Single Flux Quantum Logic (http://pavel.physics.sunysb.edu/RSFQ/Research/WhatIs/rsfqre2m.html)
[9]. Proceedings of Applied Superconductivity Confence 2016, plenary talk by N.Yoshikawa, Low-energy high-performance computing based on superconducting technology (http://ieeecsc.org/pages/plenary-series-applied-superconductivity-conference-2016-asc-2016#Plenary7)
[10]. A.Y.Herr and Q.P.Herr, Josephson magnetic random access memory system and method, International patent nr:8 270 209 B2, 2012
[11]. J.A.Blackburn, M.Cirillo, N.Gronbech-Jensen, A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures, arXiv:1602.05316v1, 2016
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.
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
Minimal M-gs Open and Maximal M-gs Closed Sets In Interior Minimal Spaceinventionjournals
The main objective of this paper is to study the notions of Minimal M-GS Closed set, Maximal M-GS
Open set, Minimal M-GS Open set and Maximal M-GS Closed set and their basic properties in Interior Minimal
Space.
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.
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Field Induced Josephson Junction (FIJJ) is defined as the physical system made by placement of ferromagnetic strip directly or indirectly [insulator layer in-between] on the top of superconducting strip [3, 4, 7]. The analysis conducted in extended Ginzburg-Landau, Bogoliubov-de Gennes and RCSJ [11] models essentially points that the system is in most case a weak-link Josephson junction [2] and sometimes has features of tunneling Josephson junction [1]. Generalization of Field Induced Josephson junctions leads to the case of network of robust coupled field induced Josephson junctions [4] that interact in inductive way. Also the scheme of superconducting Random Access Memory (RAM) for Rapid Single Flux [8, 9] quantum (RSFQ) computer is drawn [6, 10] using the concept of tunneling Josephson junction [1] and Field Induced Josephson junction [3, 4].
The given presentation is also available by YouTube (https://www.youtube.com/watch?v=uIqXqiwDsSM).
Literature
[1]. B.D.Josephson, Possible new effects in superconductive tunnelling, PL, Vol.1, No. 251, 1962
[2]. K.Likharev, Josephson junctions Superconducting weak links, RMP, Vol. 51, No. 101, 1979
[3]. K.Pomorski and P.Prokopow, Possible existence of field induced Josephson junctions, PSS B, Vol.249, No.9, 2012
[4]. K.Pomorski, PhD thesis: Physical description of unconventional Josephson junction, Jagiellonian University, 2015
[4]. K.Pomorski, H.Akaike, A.Fujimaki, Towards robust coupled field induced Josephson junctions, arxiv:1607.05013, 2016
[6]. K.Pomorski, H.Akaike, A.Fujimaki, Relaxation method in description of RAM memory cell in RSFQ computer, Procedings of Applied Conference 2016 (in progress)
[7]. J.Gelhausen and M.Eschrig, Theory of a weak-link superconductor-ferromagnet Josephson structure, PRB, Vol.94, 2016
[8]. K.K. Likharev, Rapid Single Flux Quantum Logic (http://pavel.physics.sunysb.edu/RSFQ/Research/WhatIs/rsfqre2m.html)
[9]. Proceedings of Applied Superconductivity Confence 2016, plenary talk by N.Yoshikawa, Low-energy high-performance computing based on superconducting technology (http://ieeecsc.org/pages/plenary-series-applied-superconductivity-conference-2016-asc-2016#Plenary7)
[10]. A.Y.Herr and Q.P.Herr, Josephson magnetic random access memory system and method, International patent nr:8 270 209 B2, 2012
[11]. J.A.Blackburn, M.Cirillo, N.Gronbech-Jensen, A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures, arXiv:1602.05316v1, 2016
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.
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
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.
Rough set theory is a powerful tool to analysis the uncertain and imprecise problem in information systems. Also the soft set and lattice theory can be used as a general mathematical tool for dealing with uncertainty. In this paper, we present a new concept, soft rough lattice where the lower and upper approximations are the sub lattices and narrate some properties of soft rough lattice with some examples. Payoja Mohanty "Soft Lattice in Approximation Space" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-6 | Issue-6 , October 2022, URL: https://www.ijtsrd.com/papers/ijtsrd52246.pdf Paper URL: https://www.ijtsrd.com/other-scientific-research-area/other/52246/soft-lattice-in-approximation-space/payoja-mohanty
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.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
In this paper we introduce the concept of connectedness in fuzzy rough topological spaces.
We also investigate some properties of connectedness in fuzzy rough topological spaces.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
UiPath Test Automation using UiPath Test Suite series, part 4
Stability criterion of periodic oscillations in a (7)
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.7, 2013
32
On Nano Continuity
M. Lellis Thivagar1 *
Carmel Richard 2
1. School of Mathematics, Madurai Kamaraj University,Madurai- 625021, Tamilnad, India
2. Department of Mathematics, Lady Doak College, Madurai- 625002, Tamilnad, India
* E-mail of the corresponding author: carmel09richard@gmail.com
Abstract
The purpose of this paper to propose a new class of functions called nano continuous functions and derive their
characterizations in terms of nano closed sets, nano closure and nano interior. There is also an attempt to define
nano-open maps, nano closed maps and nano homeomorphism
Keywords: Nano topology, nano-open sets, nano closed sets, nano interior, nano closure, nano continuous
functions, nano-open maps, nano closed maps, nano homeomorphism
2010 AMS Subject Classification:54B05,54C05
1. Introduction
Continuity of functions is one of the core concepts of topology. In general, a continuous function is one, for
which small changes in the input result in small changes in the output. The notion of Nano topology was
introduced by Lellis Thivagar [3], which was defined in terms of approximations and boundary region of a
subset of an universe using an equivalence relation on it. He has also defined nano closed sets, nano-interior and
nano closure. In this paper we have introduced a new class of functions on nanotopological spaces called nano
continuous functions and derived their charaterizations in terms of nano closed sets, nano closure and nano
interior. We have also established nano-open maps, nanoclosed maps and nano homeomorphisms and their
representations in terms of nano closure and nano interior.
2. Preliminaries
Definition 2.1 [5]: Let U be a non-empty finite set of objects called the universe and R be an equivalence
relation on U named as the indiscernibility relation.Then U is divided into disjoint equivalence classes.
Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U , R)
is said to be the approximation space. Let X ⊆ U .
(i) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as
X with respect to R and it is denoted by LR (X) . That is, LR (X) = })(:)({ XxRxR
x
⊆
∈
UU
, where R(x) denotes the
equivalence class determined by x ∈ U.
(ii) The upper approximation of X with respect to R is the set of all objects, which can be possibly classified
as X with respect to R and it is denoted by UR(X). That is, UR(X) = })(:)({ φ≠∩
∈
XxRxR
x
UU
(iii) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor
as not-X with respect to R and it is denoted by BR(X) . That is, BR(X) = UR(X) - LR (X).
Property 2.2 [5]: If ( U, R) is an approximation space and X, Y ⊆ U , then
i) LR (X) ⊆ X ⊆ UR(X)
ii) LR (ϕ ) = UR (ϕ) = ϕ
iii) LR(U) = UR(U) = U
iv) UR ( X ∪ Y ) = UR ( X ) ∪ UR ( Y )
v) UR ( X ∩ Y ) ⊆ UR ( X ) ∩ UR ( Y )
vi) LR ( X ∪ Y ) ⊇ LR ( X ) ∪ LR ( Y )
vii) LR ( X ∩ Y ) = LR ( X ) ∩LR ( Y )
viii) LR (X) ⊆ LR (Y ) and UR (X) ⊆ UR (Y ) whenever X ⊆ Y
ix) UR ( X C
) = [LR ( X ) ] C
and LR ( X C
) = [UR ( X ) ] C
x) UR( UR (X) ) = LR (UR ( X ) ) = UR ( X )
xi) LR (LR (X) ) = UR (LR (X) ) = LR ( X )
Definition 2.3 [3]: Let U be a non-empty, finite universe of objects and R be an equivalence relation on U. Let
X ⊆ U. Let )(XRτ = { U, ϕ, LR ( X ), UR ( X ), BR(X) }. Then )(XRτ is a topology on U , called as the nano
topology with respect to X. Elements of the nano topology are known as the nano-open sets in U and ( U,
)(XRτ ) is called the nano topological space. )]([ XRτ C
is called as the dual nano topology of )(XRτ .
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Elements of )]([ XRτ C
are called as nano closed sets.
Remark 2.4 [3]: The basis for the nano topology )(XRτ with respect to X is given by =)(XRβ { U, LR
( X ) ,B R(X) }.
Definition 2.5 [3] If ( U, )(XRτ ) is a nano topological space with respect to X where X ⊆ U and if A ⊆ U,
then the nano interior of A is defined as the union of all nano-open subsets of A and it is denoted by NInt(A).
That is, NInt(A) is the largest nano-open subset of A. The nano closure of A is defined as the intersection of all
nano closed sets containing A and it is denoted by NCl(A). That is, NCl(A) is the smallest nano closed set
containing A.
Remark 2.6 :Throughout this paper, U and V are non-empty, finite universes; X ⊆ U and Y ⊆ V ; U/R and V/R’
denote the families of equivalence classes by equivalence relations R and R’ respetively on U and V. ( U,
)(XRτ ) and ( V, )(' YRτ ) are the nano topological spaces with respect to X and Y respectively.
3.Nano continuity
Definition 3.1 : Let ( U, )(XRτ ) and ( V, )(' YRτ ) be nano topological spaces. Then a mapping f : ( U,
)(XRτ ) →( V, )(' YRτ ) is nano continuous on U if the inverse image of every nano-open set in V is nano-
open in U.
Example 3.2 Let U = },,,{ dcba with U/R= }}{},{},,{{ dbca . Let ⊆},{= daX U. Then
}},{},,,{},{,,{=)( cadcadXR φτ U . Let },,,{= wzyxV with }}{},,{},{{=/ wzyxR'
V and
},{= zxY . Then }},{},,,{},{,,{=)( zyzyxxYR' φτ V . Define f: U → V f as f(a) = y, f(b) = w, f(c) = z,
f(d) = x. Then },,{=}),,({},{=})({ 11
dcazyxfdxf −−
and },{=}),({1
cazyf −
. That is, the inverse
image of every nano-open set in V is nano -open in U. Therefore, f is nano continuous.
The following theorem characterizes nano continuous functions in terms of nano closed sets.
Theorem 3.3 : A function f : ( U, )(XRτ ) →( V, )(' YRτ ) is nano continuous if and only if the inverse image
of every nano closed set in V is nano closed in U.
Proof: Let f be nano continuous and F be nano closed in V. That is, V - F is nano-open in V Since f is nano
continuous, )(1
Ff −−
V is nano-open in U. That is, U - )(1
Ff −
is nano-open in U. Therefore, )(1
Ff −
is
nano closed in U. Thus, the inverse image of every nano closed set in V is nano closed in U , if f is nano
continuous on U. Conversely, let the inverse image of every nano closed set be nano closed. Let G be nano-open
in V. Then V - G is nano closed in V. Then, )(1
Gf −−
V is nano closed in U. That is,
U - )(1
Gf −
is nano closed in U. Therefore, )(1
Gf −
is nano-open in U . Thus,the inverse image of every
nano-open set in V is nano-open in U. That is, f is nano continuous on U.
In the following theorem, we establish a characterization of nano continuous functions in terms of nano closure.
Theorem 3.4 : A function f : ( U, )(XRτ ) →( V, )(' YRτ ) is nano continuous if and only if
))(())(N( AfClAClf N⊆ for every subset A of U.
Proof: Let f be nano continuous and U⊆A . Then V⊆)(Af . NCl(f(A)) is nano closed in V . Since f is
nano continuous, )))(((1
AfClf N−
is nano closed in U . Since ))(()( AfClAf N⊆ ,
)))(((1
AfClfA N−
⊆ . Thus ))(((1
AfClf N−
is a nano closed set containing A. But, N Cl(A) is the
smallest nano closed set containing A. Therefore ))(()( 1
AClfACl NN −
⊆ . That is ,
))(())(( AfClAClf NN ⊆ . Conversely, let f(NCl(A)) ⊆ NCl(f(A)) for every subset A of U. If F is nano
closed in V, since f−1
(F) ⊆ U, f(NCl(f−1
(F))) ⊆ NCl(f(f−1
(F))) ⊆ NCl(F).
That is, NCl(f−1
(F)) ⊆ f−1
(NCl(F)) = f−1
(F)) , since F is nano closed. Thus ).()(( 11
FfFfCl −−
⊆N But
))(()( 11
FfClFf −−
⊆ N . Therefore, )(=))(( 11
FfFfCl −−
N . Therefore, )(1
Ff −
is nano closed in U
for every nano closed set F in V . That is, f is nano continuous.
Remark 3.5 : If f : ( U, )(XRτ ) →( V, )(' YRτ ) is nano continuous, then ))(( AClf N is not necessarily
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equal to ))(( AfClN where A ⊆ U. For example, let },,,{= dcbaU ; }}{},,{},{{=/ cdbaRU . Let
},,{= dcaX . Then }},{},,{,,{=)( dbcaXR φτ U . Let },,,{= wzyxV with
}}{},{},,{{=/ wyzxR′V . Let },{= yxY . Then }},{},,,{},{,,{=)( zxzyxyYR φτ V′ . Let
))(,())(,(: YXf RR ′→ ττ VU be given by f(a) = y, f(b) = x, f(c) = y, f(d) = x. Then
φφ =)(,=)( 11 −−
ff UV , U=}),,({},,{=})({ 11
zyxfcayf −−
and },{=}),({1
dbzxf −
. That is, the
inverse image of every nano-open set in V is nano-open in U. Therefore, f is nano continuous on U. Let
V⊆},{= caA . Then }{=}),({=))(( ycafAClf N . But, },{=})({=))(( wyyClAfCl NN . Thus,
))(()(( AfClAClf NN ≠ , even though f is nano continuous. That is, equality does not hold in the previous
theorem when f is nano continuous.
Theorem 3.6 : Let ))(,( XRτU and ))(,( YR′τV be two nanotopological spaces where U⊆X and V⊆Y .
Then ),(),(,,{=)( YUYLY RRR ′′′ φτ V )}(YBR′ and its basis is given by )}(),(,{= YBYL RRR ′′′ VB . A
function ))(,())(,(: YXf RR ′→ ττ VU is nano continuous if and only if the inverse image of every member
of R′B is nano-open in U.
Proof: Let f be nano continuous on U. Let RB ′∈B . Then B is nano-open in V . That is, )(YB R′∈τ . Since f
is nano continuous, )()(1
XBf Rτ∈−
. That is,the invese image of every member of R′B is nano-open in U.
Conversely, let the inverse image of every member of R′B be nano-open in U. Let G be a nano-open in V .
Then }:{= 1B∈BBG U , where R′⊂BB1 . Then
}:)({=}):{(=)( 1
1
1
11
BB ∈∈ −−−
BBfBBfGf UU , where each )(1
Bf −
is nano-open in U and
hence their union, which is )(1
Gf −
is nano-open in U. Thus f is nano continuous on U.
The above theorem characterizes nano continuous functions in terms of basis elements. In the following
theorem, we characterize nano continuous functions in terms of inverse image of nano closure.
Theorem 3.7 : A function ))(,())(,(: YXf RR ′→ ττ VU is nano continuous if and only if
))(())(( 11
BClfBfCl NN −−
⊆ for every subset B of V .
Proof: If f is nano continuous and V⊆B , )(BClN is nano closed in V and hence ))((1
BClf N−
is nano
closed in U . Therefore, ))((=))](([ 11
BClfBClfCl NNN −−
. Since )(BClB N⊆ ,
))(()( 11
BClfBf N−−
⊆ . Therefore, ))((=)))((())(( 111
BClfBClfClBfCl NNNN −−−
⊆ . That is,
))(())(( 11
BClfBfCl NN −−
⊆ . Conversely, let ))(())(( 11
BClfBfCl NN −−
⊆ for every V⊆B . Let B
be nanoclosed in V . Then BBCl =)(N . By assumption, )(=))(()( 111
BfBClfBClf −−−
⊆ NN . Thus,
)()( 11
BfBClf −−
⊆N . But ))(()( 11
BfClBf −−
⊆ N . Therefore, )(=))(( 11
BfBfCl −−
N . That is,
)(1
Bf −
is nano closed in U for every nano closed set B in V . Therefore, f is nano continuous on U.
The following thereom establishes a criteria for nano continuous functions in terms of inverse image of nano
interior of a subset of V .
Theorem 3.8 : A function ))(,())(,(: YXf RR ′→ ττ VU is nano continuous on U if and only if
))(())(( 11
BfIntBIntf −−
⊆ NN for every subset B of V .
Proof: Let f be nano continuous and V⊆B . Then )(BIntN is nano-open in ))(,( YR′τV . Therefore
))((1
BIntf N−
is nano-open in ))(,( XRτU . That is, ))](([=))(( 11
BIntfIntBIntf NNN −−
. Also,
BBInt ⊆)(N implies that )())(( 11
BfBIntf −−
⊆N . Therefore
))(())](([ 11
BfIntBIntfInt −−
⊆ NNN . That is, ))(())(( 11
BfIntBIntf −−
⊆ NN . Conversely, let
))(())(( 11
BfIntBIntf −−
⊆ NN for every subset B of V . If B is nano-open in V , BBInt =)(N . Also,
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))(())(( 11
BfIntBIntf −−
⊆ NN . That is, ))(()( 11
BfIntBf −−
⊆ N . But )())(( 11
BfBfInt −−
⊆N .
Therefore, ))((=)( 11
BfIntBf −−
N . Thus, )(1
Bf −
is nano-open in U for every nano-open set B in V .
Therefore, f is nano continuous.
Example 3.9 : Let },,,{= dcbaU with }}{},{},,{{=/ cbdaRU . Let U⊆},{= caX . Then the
nanotopology, )(XRτ with respect to X is given by }},{},,,{},{,,{ dadcacφU and hence the nanoclosed
sets in U are }{},,,{,, bdbaφU and },{ cb . Let },,,{= wzyxV with R′/V }}{},{},{},{{= wzyx . Let
V⊆},{= wxY . Then the nanotopology on V with respect to Y is given by }},{,,{=)( wxYR φτ V′ , and
the nanoclosed sets in V are φ,V and },{ zy . Define VU →:f as xaf =)( , ybf =)( , zcf =)( and
wdf =)( . Then f is nano continuous on U, since inverse image of every nano-open set in V is nano-open in
U . Let V⊂}{= yB . Then },{=}),({=))(( 11
cbzyfBClf −−
N and }{=))(( 1
bBfCl −
N . Thus,
))(( 1
BfCl −
N ≠ ))((1
BClf N−
. Also when V⊆},,{= wzxA ,
},{=}),({=))(( 11
dawxfAIntf −−
N but },,{=}),,({=))(( 1
dcadcaIntAfInt NN −
. That is,
))(())(( 11
AfIntAIntf −−
≠ NN . Thus, equality does not hold in theorems 5.6 and 5.7 when f is nano
continuous.
Theorem 3.10 : If ))(,( XRτU and ))(,( YR′τV are nano topological spaces with respect to U⊆X and
V⊆Y respectively, then for any function VU →:f , the following are equivalent:
1. f is nano continuous.
2. The inverse image of every nano closed set in V is nano closed in U.
3. ))(())(( AfClAClf NN ⊂ for every subset A of V .
4. The inverse image of every member of the basis BR ’ of )(' YR
τ is nano-open in U.
5. ))(())(( 11
BClfBfCl NN −−
⊆ for every subset B of V .
6. ))(())(( 11
BfIntBIntf −−
⊂ NN for every subset B of V .
Proof of the theorem follows from theorems 3.3 to 3.8.
Definition 3.11 : A subset A of a nanotopological space ))(,( XRτU is said to be nano dense if UN =)(ACl .
Remark 3.12 : Since UN =)(XCl is rough topological space ))(,( XRτU with respect to X where U⊂X ,
X is nano dense in U.
Example 3.13 : Let },,,{= dcbaU with }}{},,{},{{=/ dcbaRU . Let },{= caX . Then
}},{},,,{},{,,{=)( cbcbaaXR φτ U and the rough closed sets in U are }{},,,{,, ddcbφU and },{ da . If
},,,{= wzyxV with }},{},{},{{=/ wzyxR′V and },{= zxY , then
}},{},,,{},{,,{=)( wzwzxxYR φτ V′ and the nanoclosed sets in V are },{},{},,,{,, yxywzyφV . Define
a function VU →:f as wcfzbfzaf =)(,=)(,=)( and ydf =)( . Then f is nano continuous since
the inverse image of every nano-open set in V is nano-open in U. Let U⊆},,{= dbaA . UN =)(ACl and
hence A is nano dense in U. But VNN ≠},,{=}),({=)( wzyzyClAClf . Therefore, f(A) is not nano
dene even though A is nano dense and f is nano continuous. Thus, a nano continuous function does not map nano
dense sets into nano dense sets.
In the following theorem, we establish that a nano continous function maps nano dense sets into nano dense sets,
provided it is onto.
Theorem 3.14 : Let ))(,())(,(: YRXRf ′→ ττ VU be an onto, nano continuous function. If A is nano dense in U ,
then )(Af is nano dense in V .
Proof: Since A is nano dense in UNU =)(, ACl . Then VUN =)(=))(( fAClf , since f is onto. Since f is
nano continuous on ))(())((, AfClAClf NNU ⊆ . Therefore, ))(( AfClNV ⊆ . But VN ⊆))(( AfCl .
Therefore, VN =))(( AfCl . That is, f(A) is nano dense in V . Thus, a nano continuous function maps nano
dense sets into nano dense sets, provided it is onto.
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4. Nano-open maps, Nano closed maps and Nano homeomorphism
Definition 4.1 : A function ))(,())(,(: YXf RR ′→ ττ VU is a nano-open map if the image of every nano-
open set in U is nano-open in V . The mapping f is said to be a nanoclosed map if the image of every
nanoclosed set in U is nanoclosed in V .
Theorem 4.2 : A mapping ))(,())(,(: YXf RR ′→ ττ VU is nanoclosed map if and only if
))(())(( AClfAfCl NN ⊆ , for every subset A of U.
Proof: If f is nanoclosed, ))(( AClf N is nanoclosed in V , since )(AClN is nano closed in U. Since
)(AClA N⊆ , ))(()( AClfAf N⊆ .Thus ))(( AClf N is a nano closed set containing f(A). Therefore,
))(())(( AClfAfCl NN ⊆ . Conversely, if ))(())(( AClfAfCl NN ⊆ for every subset A of U and if F
is nanoclosed in U, then FFCl =)(N and hence )(=))(())(()( FfFClfFfClFf NN ⊆⊆ . Thus,
))((=)( FfClFf N . That is, )(Ff is nanoclosed in V . Therefore, f is a nanoclosed map.
Theorem 4.3 : A mapping ))(,())(,(: YXf RR ′→ ττ VU is nano-open map if and only if
))(())(( AfIntAIntf NN ⊆ , for every subset U⊆A .
Proof is similar to that of theorem 4.2
Definition 4.4 : A function ))(,())(,(: YXf RR ′→ ττ VU is said to be a nano homeomorphism if
1. f is 1-1 and onto
2. f is nano continuous and
3. f is nano-open
Theorem 4.5 : Let ))(,())(,(: YXf RR ′→ ττ VU be a one-one onto mapping. Then f is a nano
homeomorphism if and only if f is nano closed and nanocontinuous.
Proof: Let f be a nano homeomorphism. Then f is nano continuous. Let F be an arbitrary nano closed set in
))(,( XRτU . Then U -F is nano-open. Since f is nano- open, )( Ff −U is nano- open in V . That is,
)(Ff−V is nano-open in V . Therefore, )(Ff is nano-closed in V . Thus, the image of every nano closed
set in U is nano closed in V . That is, f is nano closed. Conversely, let f be nano closed and nano continuous.
Let G be nano-open in ))(,( XRτU . Then U -G is nano closed in U . Since f is nano closed,
)(=)( GfGf −− VU is nano closed in V . Therefore, )(Gf is nano-open in V . Thus, f is nano-open and
hence f is a nano homeomorphism.
The following theorem provides a condition on a nano continuous function under which equality holds
in theorem 3.3
Theorem 4.6 : A one-one map f of ))(,( XRτU onto ))(,( YR′τV is a nano homeomorphism iff
)]([=))(( AfClAClf NN for every subset A of U.
Proof: If f is a nano homeomorphism, f is nano continuous and nano closed. If U⊆A ,
))(())(( AfClAClf NN ⊂ , since f is nano continuous. Since )(AClN is nano closed in U and f is nano
closed, ))(( AClf N is nano closed in V . ))((=)))((( AClfAClfCl NNN . Since
))(()(),( AClfAfAClA NN ⊆⊆ and hence ))((=))](([))(( AClfAClfClAfCl NNNN ⊆ .
Therefore, ))(())(( AClfAfCl NN ⊆ . Thus, ))((=))(( AfClAClf NN if f is a nano homeomorphism.
Conversely, if ))((=))(( AfClAClf NN for every subset A of U, then f is nano continuous . If A is nano
closed in AACl =)(,NU which implies )(=))(( AfAClf N . Therefore, )(=))(( AfAfClN . Thus, f(A)
is nano closed in V , for every nano closed set A in U. That is f is nano closed. Also f is nano continuous. Thus,
f is a nano homeomorphism.
5. Application
In this section we apply the concept of nano continuous functions in a day-to-day problem.
Consider the cost of a cab ride as a function of distance travelled. Let },,,,,{= 654321 xxxxxxU be the
universe of distances of six different places from a railway junction and let },,,,,{= fedcbaV be the
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universe of cab fares to reach the six destinations in U from the railway junction . We know that the fares
depend on the distance of the places.Let }}{},,{},,{},{{=/ 653421 xxxxxxRU and let },,{= 321 xxxX , a
subset of U .Then the nano topology on U is given by
}},,,{},,,,,{},{,,{=)( 5432543211 xxxxxxxxxxXR φτ U . Let }}{},,{},,{},{{=/ fecdbaR′V and let
Y = },,{ cba . Then the nano topology )(YR′τ on V with respect to Y is given by
}},,,{},,,,,{},{,,{ edcbedcbaaφV . Define VU →:f as ,=)(,=)(,=)( 321 cxfbxfaxf
.=)(,=)(,=)(, 654 fxfexfdxf Then },{=})({,=)(,=)( 1
111
xafff −−−
φφVU
},,,,{=}),,,,({ 54321
1
xxxxxedcbaf −
and },,,{=}),,,({ 5432
1
xxxxedcbf −
. That is, the inverse
image of every nano-open set in V is nano-open in U. Therefore f is nano continuous.Also we note that the
image of every nano-open set in U is nano-open in V and f is a bijection . Thus, f is a nano homeomorphism.
Therefore, the cost of a cab ride, as a function of distance travelled, is a nano homeomorphism.
6. Conclusion
The theory of nano continuous functions has a wide variety of applications in real life. In this paper, we have
shown that the cost of cab rides, as a function of distance travelled, is not only a nano continuous function but
also a nano homeophormism.Similarly, nano continuous functions have a wide range of applications such as
growth of a plant over time, depreciaton of machine and temperature at various times of the day.Thus, nano
continuous functions, in near future, can be applied to more day-to-day situations.
References
Lashin E.F., Kozae A.M., Abo Khadra A.A. & Medhat T. (2005), ``Rough set theory for topological spaces'',
International Journal of Approximate Reasoning, 40/1-2, 35-43,
Lashin E.F. & Medhat T. (2005) ``Topological reduction of information systems'', Chaos, Solitons and Fractals,
25 277-286,.
Lellis Thivagar M. & Carmel Richard, ``Note on Nanotopological spaces'' (Communicated).
Pawlak Z. (1982) , ``Rough sets'', International Journal of Information and Computer Science, 11(5); 341-356,.
Pawlak Z. (1991) , ``Rough sets - Theoretical Aspects of Reasoning about data'', Kluwer Academic Publishers,
Dordrecht, Boston, London,.
Rady E.A. ( 2004) , Kozae A.M. and Abd El-Monsef M.M.E., ``Generalized Rough Sets'', Chaos, Solitons and
Fractals, 21, 49-53.
Salama S. (2011) , ``Some topological properties of rough sets with tools for data mining'', International Journal
of Computer Science Issues, Vol.8, Issue 3, No.2, 588-595.
Skowron A. (1988) , ``On Topology in Information System'', Bull. Polish Acad. Sci., Math., 36/7-8, 477-479,.
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