View of CONCEPTUAL STUDY BASED ON OPEN SETS IN IDEAL TOPOLOGICAL SPACES

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) (ISSN NO. 2456-1037) Vol. 03, Conference (IC-RASEM) Special Issue 01, September 2018 Available Online: www.ajeee.co.in/index.php/AJEEE

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CONCEPTUAL STUDY BASED ON OPEN SETS IN IDEAL TOPOLOGICAL SPACES Dr. H. K. Tripathi

Lecturer, Govt. Women’s Polytechnic College, Jabalpur-482001

Abstract- In Algebraic Topology we understand the topological aspects of surface like objects that arise by combining elementary shapes, such as polygons or polyhedral. The German geometers August Mobius and Felix Klein published works on ―one-sided‖ surfaces in 1853 and 1882 respectively. Mobius band example, now known as the Mobius strip, may be constructed by gluing together the ends of a long rectangular strip of paper that has been given a half twist. Surfaces containing subsets homeomorphism to the Mobius strip are called non-oriental surfaces and play an important role in the classification of two- dimensional surfaces. Klein provided an example of a one-sided surface that is closed, that is, without any one-dimensional boundaries. This example, now called the Klein bottle, cannot exist in three-dimensional space without intersecting itself and therefore this surface was interesting to mathematicians who previously had considered surfaces only in three-dimensional space.

Keywords: Open set, Topological spaces.

1 INTRODUCTION

The motivating insight behind topology is geometry, because some geometric problems depend not on the exact shape of objects involved but rather on the way they are put together. Topology is one such branch of classical mathematics which has penetrated into several domains of mathematics and has stimulated their growth. The influence of topology can be felt on many branches of applied and pure mathematics especially on Geometry and Analysis. Study of topology in the last century convinced topologists that topology can be described as certain axioms on the collection of subsets of anon-empty set 𝑋.

The concept of topological spaces are generalization of the concept of Metric Spaces. In a Metric Space 𝑋 we study the distance function 𝑑: 𝑋 × 𝑋 → 𝑅 and this distance function 𝑑 gives a Basis for a topology on 𝑋. Thus by generalize a topology by the Basis set we get a topology on X. Hence every Metric space we can regard as a topological space.

However there are several topological space which are non-metrizable spaces, i.e., there is no metric which induces the topology of the topological spaces.

1.1 Semi-Open sets and its Properties Definition1.1.1 [1]: Let (𝑋, 𝜏) be a topological space and let A ⊆ X. Then𝐴 is said to be semi-open set if A ⊆ 𝐶𝑙 𝐼𝑛𝑡 𝐴 .

Remark 1.1.1: In a topological space(𝑋, 𝜏) the empty set ∅ and whole set X are semi- open sets.

Example 1.1.1: Let 𝑋 = 𝑥₁, 𝑥₂, 𝑥₃ be the topological space with respect to topology 𝜏 = ∅, 𝑋, 𝑥2 , 𝑥₂, 𝑥3 on X. Let us consider the set A= 𝑥1, 𝑥2 in X. We see that Int(A)

= {𝑥2} and 𝐶𝑙 𝐼𝑛𝑡 𝐴 = 𝑋. Therefore A ⊆ 𝐶𝑙 𝐼𝑛𝑡 𝐴 . Thus the set A is semi- open set in X.

Proposition1.1.1 [1]: In a topological space (𝑋, 𝜏) each open set is semi-open but not the converse.

Remark 1.1.2: In a topological space (𝑋, 𝜏) if A is a non-empty semi-open set in X then 𝐼𝑛𝑡 𝐴 is also non-empty.

1.2 Applications of General Topology Topology is one such branch of classical mathematics which has penetrated into several domains of mathematics and has stimulated their growth. The influence of topology can be felt on many branches of applied and pure mathematics especially on Geometry and Analysis. Topology has immense applications in Physics, Mechanics, Robotics, Biology, Computer Science and even Humanities. Topology is extensively used in data mining, geometric design, molecular design [2], engineering design [3], digital topology, computer graphics, information system [4], particle physics, quantum physics[2] etc. In fact, the success of topology is due in part to the development it fostered in related parts of mathematics ranging from answers to questions in the foundations of analysis to the study of the fundamental group of knots, homology groups and manifolds.

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2 1.2.1 Application of Topology in Physics According to normal Howes-uniform structure are the most important structure from the Physicist’s vision. The importance of uniform spaces from the physicist’s points of view is also well brought out by the proceeding of the Nashville Topological conference. In fact, topology has intrigued Particle Physicists for a long time. Recall that Donaldson used the Yand Mills fields equations of Mathematical Physics, themselves generalization of Maxwell’s equations to study in 4- space , there by reversing tradition by applying methods from physics to the understanding of Topology.

1.2.2 Application of Topology in Engineering

In recent year, it has been reported that a crucial problem of some electric industry was not solved by its R and D department whereas the topologist s solve it by using some set theoretic topological approach . Daniel R. Barker has established that topological techniques are used in several robotics applications. Topology has been applied to production and distribution problems.

2 GENERALIZED FORMS OF g-OPEN AND g-CLOSED SETS

2.1 Introduction

In 2005, Csaszar [5] has generalized the notions of semi-open set, pre open set,



-open set and



-open set in the category of generalized topological spaces. There are four fundamental concepts of generalized open sets in Topological spaces. These generalized open sets are known as semi-open sets, pre-open sets,



-open sets and



-open sets. Maitra and Tripathi [6] have obtained significant properties of interior and closure of a set in generalized topological spaces. In this chapter we have studied g-semi-open, g- pre-open, g-𝛼-open, g-𝛽-open and corresponding generalized forms of g- closed sets.

2.2 g-Semi-Open Sets

In this section we have obtained significant properties of g-semi open sets.

Further we have constructed some useful examples.

Definition 2.2.1 [6]: Let (𝑋, 𝜏_𝑔)be a generalized topological space and let 𝐴 ⊆

𝑋. Then the set A is said to be g-semi- open set, if 𝐴 ⊆ 𝑐_𝑔 𝑖_𝑔 𝐴 .

Remark: Ingeneralized topological space 𝑋, 𝜏𝑔 the empty set 𝜙 and whole set 𝑋 are always g-semi-open sets.

Proposition 2.2.1: Let 𝑋, 𝜏𝑔 be a generalized topological space. If A is a g- open set in X then A is g-semi-open set.

Proof: Let X be a generalized topological space and A



X. Suppose A is a g-open set in X. Then 𝑖𝑔(𝐴) = 𝐴. Since 𝐴



𝑐𝑔 𝐴 , we have, 𝐴 ⊆ 𝑐𝑔 𝑖𝑔 𝐴 . Hence A is g- semi-open set in X.

However the converse of above Proposition 2.3.2 is not necessarily true.

In the following example we see that A is a g-semi-open set but A is not a g-open set in X.

Example 2.2.1: Let 𝑋 = { 𝑎, 𝑏, 𝑐, 𝑑 } and let us consider generalized topology 𝜏𝑔 = 𝜙, 𝑋, 𝑎, 𝑏 , 𝑏, 𝑑 , {𝑎, 𝑏, 𝑑} on X. Suppose 𝐴 = {𝑎, 𝑏, 𝑐 }. Then we see that A is a g- semi-open set in X but not g-open set in X.

Remark 2.2.1: Ina generalized topological space 𝑋, 𝜏𝑔 if A is non empty g-semi-open subset of X then 𝑖_𝑔 𝐴 is also a non-empty subset of X.

Proposition 2.2.2: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A

⊆ 𝑋. Then A is g-semi-open set if 𝑐𝑔 𝐴 = 𝑐𝑔 𝑖𝑔 𝐴 .

Proof: Let A be a g-semi-open set in X . Then we have 𝐴 ⊆ 𝑐_𝑔 𝑖_𝑔 𝐴 . This implies𝑐𝑔 𝐴 ⊆ 𝑐𝑔(𝑐𝑔 𝑖𝑔 𝐴 =𝑐𝑔 𝑖𝑔 𝐴 , i.e., 𝑐𝑔 𝐴 𝑐_𝑔 𝑖𝑔 𝐴 . Since 𝑖𝑔 𝐴 ⊆ 𝐴 , we have 𝑐𝑔 𝑖𝑔 𝐴 ⊆ 𝑐_𝑔 𝐴 . Hence we find that 𝑐_𝑔 𝐴 𝑐_𝑔 𝑖_𝑔 𝐴 .

Conversely, suppose that 𝑐𝑔 𝐴 𝑐𝑔 𝑖𝑔 𝐴 . Since 𝐴 ⊆ 𝑐𝑔 𝐴 we have 𝐴 ⊆ 𝑐_𝑔 𝑖_𝑔 𝐴 . Thus A is g-semi-open set in X.

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3 2.3 g-Semi-Closed Sets:

Definition 2.3.1 [6]: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A

⊆ 𝑋. Then the set A is said to be g-semi- closed set if (X - A) is g-semi-open set in 𝑋.

Remark 2.3.1: In a generalized topological space 𝑋, 𝜏_𝑔 the empty set 𝜙 and whole set 𝑋 are always g-semi-closed sets.

Proposition 2.3.1: Let 𝑋, 𝜏𝑔 be a generalized topological space. If A is a g- closed set in X then A is g-semi-closed set.

Proof: Let X be a generalized topological space and let A be a g-closed set in X.

Then 𝑐𝑔(𝐴) = 𝐴. Now we have, 𝑐𝑔(𝑖𝑔(𝑋 − 𝐴)) = 𝑐_𝑔(𝑋 − 𝑐_𝑔(𝐴)) = 𝑋 − 𝑖_𝑔(𝑐_𝑔(𝐴)) = 𝑋 − 𝑖𝑔(𝐴)



(𝑋 − 𝐴) (as 𝑖𝑔 𝐴 ⊆ 𝐴). Thus 𝑋 − 𝐴 ⊆ 𝑐_𝑔 𝑖𝑔 𝑋 − 𝐴 . Hence (𝑋 − 𝐴) is a g-semi-open set in X and so A is a g- semi-closed set in X.

In the following Example we see that converse of above result is not necessarily true.

Example 2.3.1: Let 𝑋 = { 𝑎, 𝑏, 𝑐, 𝑑 } and let us consider generalized topology 𝜏_𝑔 = 𝜙, 𝑋, 𝑎, 𝑏 , 𝑏, 𝑑 , {𝑎, 𝑏, 𝑑} on X. Suppose 𝐴 = {𝑑}. Then we see that A is g-semi- closed set in X but not g-closed set in X.

Proposition 2.3.2: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A



X. Then A is g-semi-closed set iff𝑖_𝑔 𝑐_𝑔 𝐴 ⊆ 𝐴.

Proof: Let A be a g-semi-closed set in X.

Then (X - A) is g-semi-open set in X. This means (𝑋 − 𝐴)



𝑐_𝑔 𝑖_𝑔 𝑋 − 𝐴 . We have, 𝑖𝑔(𝑋 − 𝐴) = 𝑋 − 𝑐𝑔(𝐴). Therefore 𝑐𝑔(𝑖𝑔(𝑋 − 𝐴) = 𝑐𝑔(𝑋 − 𝑐𝑔(𝐴)) = 𝑋 −

𝑖𝑔 𝑐𝑔 𝐴 . Thus we find that 𝑋 − 𝐴 ⊆ 𝑋 − 𝑖𝑔 𝑐𝑔 𝐴 . This implies 𝑖_𝑔 𝑐𝑔 𝐴 ⊆ 𝐴.

Conversely suppose that 𝐴 ⊆ 𝑋 and 𝑖𝑔 𝑐𝑔 𝐴 ⊆ 𝐴. Then we have 𝑋 − 𝐴 ⊆ 𝑋 − 𝑖𝑔 𝑐𝑔 𝐴 . As 𝑋 − 𝑖𝑔 𝑐𝑔 𝐴 = 𝑐_𝑔 𝑖_𝑔 𝑋 − 𝐴 , we find that 𝑋 − 𝐴 ⊆ 𝑐_𝑔 𝑖_𝑔 𝑋 − 𝐴 . Hence (𝑋 − 𝐴) is g-semi

open set and so A is g-semi-closed set in X.

Remark 2.3.2: In a generalized topological space 𝑋 if A is a g-semi-closed set and A ≠ X then 𝑐𝑔 𝐴 ≠ X .

Proposition 2.3.3: Let 𝑋, 𝜏_𝑔 be a generalized topological space and let A



X. Then A is g-semi-closed set iff 𝑖𝑔(𝐴) = 𝑖𝑔 𝑐𝑔 𝐴 .

Proof : Let A be a g-semi-closed set in X.

Then from Proposition 2.3.2, we have 𝐴 ⊇ 𝑖𝑔 𝑐𝑔 𝐴 . This implies 𝑖𝑔 (𝐴) ⊇ 𝑖𝑔 𝑖𝑔 𝑐𝑔 𝐴 = 𝑖𝑔 𝑐𝑔 𝐴 , i.e.,𝑖𝑔(𝐴) ⊇ 𝑖_𝑔 𝑐_𝑔 𝐴 . Since 𝑐_𝑔 𝐴



A, we have 𝑖_𝑔 𝑐_𝑔 𝐴



𝑖_𝑔(𝐴). Hence we have 𝑖_𝑔(𝐴) = 𝑖_𝑔 𝑐_𝑔 𝐴 .

Conversely suppose that 𝑖𝑔(𝐴)=

𝑖𝑔 𝑐𝑔 𝐴 . Since 𝑖𝑔 𝐴 ⊆ 𝐴 , we have 𝑖𝑔 𝑐𝑔 𝐴 ⊆ 𝐴. Hence from Proposition 3.3.2 it follows that A is g-semi-closed set in X .

Proposition 2.3.4: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A



X . Then A is g-semi-closed set iff there exits a g-closed set F in X such that 𝑖𝑔 F ⊆ 𝐴 ⊆ 𝐹.

Proof : Let A be a g-semi-closed set in X.

Then from Proposition 2.3.2, we have 𝑖𝑔 𝑐𝑔 𝐴 ⊆ 𝐴. Suppose 𝐹 = 𝑐𝑔 𝐴 . Then F is a g-closed set in X and 𝑖𝑔 F ⊆ 𝐴. Since 𝐴 ⊆ 𝑐_𝑔 𝐴 , we have 𝐴 ⊆ F. Hence we deduce that 𝑖𝑔 F ⊆ 𝐴 ⊆ F.

2.4 g-Pre-Open Sets:

In this section we have obtained significant properties of g-pre-open sets.

Further we have constructed some useful examples.

Definition 2.4.1[6]: Let 𝑋, 𝜏_𝑔 be a generalized topological space and let𝐴 ⊆ 𝑋.

Then the set A is said to be g-pre-open set, if 𝐴 ⊆ 𝑖𝑔 𝑐𝑔 𝐴 .

Remark 2.4.1: In a generalized topological space 𝑋, 𝜏𝑔 the empty set 𝜙

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4 and whole set 𝑋 are always g-pre-open sets.

Proposition 2.4.1: Let 𝑋, 𝜏_𝑔 be a generalized topological space. If A is a g- open set in X then A is g-pre-open set.

Proof: Let X be a generalized topological space and let A



X. Suppose A is a g- open set in X. Then 𝐴 = 𝑖𝑔(𝐴) . Since 𝐴



𝑐𝑔 𝐴 we have, 𝐴 = 𝑖𝑔 𝐴 ⊆ 𝑖𝑔 𝑐𝑔 𝐴 . Hence A is a g-pre-open set in X.

However the converse of above Proposition 2.4.1 is not necessarily true.

In the following example we see that A is a g-pre-open set but A is not g-open set in X.

Example 2.4.1: Let 𝑋 = { 𝑎, 𝑏, 𝑐, 𝑑 } and let us consider generalized topology 𝜏_𝑔 = 𝜙, 𝑋, 𝑎, 𝑏 , 𝑏, 𝑑 , {𝑎, 𝑏, 𝑑} on X. Suppose 𝐴 = {𝑎, 𝑏, 𝑐 }. Then we see that A is a g- pre-open set in X but not g-open set in X.

Theorem 2.4.1: Let (𝑋, 𝜏_𝑔) be a generalized topological space and let 𝐴 ⊆ 𝑋. Then 𝐴 is g-pre-open if and only if there exists a g-open set 𝑈 in 𝑋 such that 𝐴 ⊆ 𝑈 ⊆ 𝑐𝑔(𝐴).

Proof: Let (𝑋, 𝜏𝑔) be a generalized topological space. Suppose 𝐴 is a g-pre- open set in 𝑋. Then 𝐴 ⊆ 𝑖𝑔 𝑐𝑔 𝐴 . Let 𝑖𝑔 𝑐𝑔 𝐴 = 𝑈. Then 𝑈 is a g-open in 𝑋 and we have 𝐴 ⊆ 𝑈. Since 𝑈 = 𝑖𝑔 𝑐𝑔 𝐴 ⊆ 𝑐𝑔(𝐴). This implies 𝐴 ⊆ 𝑈 ⊆ 𝑐𝑔(𝐴).

Conversely let 𝑈be a g-open set in 𝑋 such that 𝐴 ⊆ 𝑈 ⊆ 𝑐𝑔(𝐴). This implies𝐴 ⊆ 𝑈 = 𝑖𝑔 𝑈 ⊆ 𝑖_𝑔(𝑐𝑔 𝐴 ), i.e.,𝐴 ⊆ 𝑖_𝑔 𝑐𝑔 𝐴 . Hence 𝐴 is a g-pre-open set in 𝑋.

Theorem 2.4.2: Let (𝑋, 𝜏𝑔) be a generalized topological space and let 𝐴_𝛼 _𝛼∈∧be a family of g-pre-open sets in X.

Then 𝐴 =∪_𝛼∈∧𝐴_𝛼 is g-pre-open set.

Proof: Let 𝑋 be a generalized topological space and let 𝐴_𝛼 _𝛼∈∧ be a collection of g- pre-open sets in X. Then𝐴𝛼 ⊆ 𝑖𝑔 𝑐𝑔 𝐴_𝛼 , for all 𝛼 ∈∧. Put A = ∪𝛼∈𝐽𝐴𝛼. Then we have, 𝑖𝑔 𝑐𝑔 𝐴 = 𝑖_𝑔 𝑐𝑔 ∪_𝛼∈∧𝐴𝛼 ⊇ 𝑖𝑔 ∪𝛼∈∧𝑐𝑔 𝐴_𝛼 ⊇ ∪_𝛼∈∧𝑖𝑔 𝑐𝑔 𝐴_𝛼 ⊇

∪𝛼∈∧𝐴𝛼 = 𝐴.i.e, 𝐴 ⊆ 𝑖𝑔(𝑐𝑔(𝐴)).Hence 𝐴 is a g-pre-open set in 𝑋.

In the following example we see that the intersection of two g-pre-open sets may not be g-pre-open.

Example 2.4.2: Let X = {a, b, c} and let us consider the generalized topology

_𝑔 = ∅, 𝑋, 𝑎, 𝑏 , {𝑏, 𝑐} on X. Suppose 𝐴 = {𝑎, 𝑏} and 𝐵 = {𝑎, 𝑐}. Then we see that the sets 𝐴 and 𝐵 are g-pre-open sets in X.

Now 𝐴 ∩ 𝐵 = {𝑎} and {a} is not g-pre-open set in X.

Remark 2.4.1: In the above theorem 2.4.2 it has been proved that arbitrary union of g-pre-open sets is g-pre-open set.

Further in above example 2.4.2 it is shown that intersection of two g-pre-open sets may not a g-pre-open set. Thus the collection of g-pre-open sets in a generalized topological space (𝑋, 𝜏𝑔) forms a generalized topology on 𝑋 and this collection is finer then 𝜏𝑔.

Theorem 2.4.3: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴 ⊆ 𝑋. Then A is g-pre-open set if and only if for each 𝑥 ∈ 𝐴 there exists a g-pre-open set U in X such that 𝑥 ∈ 𝑈 ⊆ 𝐴.

Proof. Let A be a g-pre-open set in X.

Then clearly for any point 𝑥 ∈ 𝐴 there exist a g-pre-open set viz., A itself satisfying the desired condition.

Conversely suppose 𝐴 ⊆ 𝑋 having the property that for each 𝑥 ∈ 𝐴 there exists a g-pre-open set 𝑈𝑥 in 𝑋 such that 𝑥 ∈ 𝑈𝑥⊆ 𝐴. Clearly we have 𝐴 = ∪𝑥∈𝐴𝑈𝑥. From Theorem 2.4.2 we note that arbitary union of g-pre-open sets is g-pre-open, therefore A is a g-pre-open set in X.

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