Research article · 2022 · Full text
International Journal of Statistics and Applied MathematicsNew Homeomorphism in topological spaces
Volume 7(6) · Pages 139–142 · DOI: 10.22271/maths.2022.v7.i6b.909 ↗
Research areasFuzzy and Soft Set TheoryFixed Point Theorems Analysis
Abstract
Author-writtenIn this paper a new class of homeomorphism called minimal Regular weakly homeomorphism and maximal Regular weakly homeomorphism are introduced and investigated and during this process some properties of the new concepts have been studied.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Research in topological spaces has previously established the study of minimal open and minimal closed sets as subclasses of open and closed sets. View evidence ↗
- The concept of weakly homeomorphism in topological spaces was introduced in the year 2000. View evidence ↗
- Minimal and maximal homeomorphisms, along with minimal open sets and maps, were studied in 2008. View evidence ↗
The question
- The paper aims to introduce and investigate a new class of homeomorphisms termed minimal Regular weakly homeomorphism and maximal Regular weakly homeomorphism. View evidence ↗
What they did
- A minimal regular weakly homeomorphism is defined as a bijective function where both the function and its inverse are minimal regular weakly continuous maps. View evidence ↗
- The author defines a min-max regular weakly homeomorphism as a bijection where both the function and its inverse are min-max regular weakly continuous maps. View evidence ↗
What they found
- Every homeomorphism is a minimal regular weakly homeomorphism, although the reverse is not necessarily true. View evidence ↗
- Every homeomorphism is a maximal regular weakly homeomorphism, but not every maximal regular weakly homeomorphism is a homeomorphism. View evidence ↗
- Minimal regular weakly homeomorphisms and maximal regular weakly homeomorphisms are independent of one another. View evidence ↗
- For a bijective and minimal regular weakly continuous function, being a minimal regular weakly homeomorphism is equivalent to being a minimal regular weakly open map or a maximal regular weakly closed map. View evidence ↗
- For a bijective and maximal regular weakly continuous function, the inverse being a maximal regular weakly homeomorphism is equivalent to the function being a maximal regular weakly open map or a minimal regular weakly closed map. View evidence ↗
- Every min-max regular weakly homeomorphism is a minimal regular weakly homeomorphism, but the converse does not hold. View evidence ↗
- Every max-min regular weakly homeomorphism is a maximal regular weakly homeomorphism, though the converse is not necessarily true. View evidence ↗
- For a bijective and min-max regular weakly continuous map, being a minimal-maximal regular weakly homeomorphism is equivalent to being a minimal-maximal regular weakly open map or a maximal-minimal regular weakly closed map. View evidence ↗
What it means
- The author concludes that the introduced classes of minimal and maximal Regular weakly homeomorphisms have been successfully investigated and their properties studied. View evidence ↗
Visual summary
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Hierarchy of Homeomorphisms
AI-generated conceptual illustration (Abstract geometric topological spaces). It explains the idea and does not show study data.
View source evidence ↗Introduction
In the year 2001 and 2003, F.Nakaoka and N.oda [1,2,3] In the year 2000, M.Sheik john [4] introduced and studied weakly homeomorphism in topological spaces and in the year 2008 B.M.Ittanagi [5] introduced and studied minimal open sets and maps in topological spaces and minimal homeomorphism and maximal homeomorphism in topological spaces. In the year 2014 R.S.Wali and Vivekananda Dembre [6,7] introduced and studied minimal weakly open sets and maximal weakly closed sets and maximal weakly open sets and minimal weakly closed sets in topological spaces.In the year 2014 [8] Vivekananda Dembre, Manjunath Gowda and Jeetendra Gurjar introduced and studied minimal weakly and maximal weakly continuous functions in topological spaces. In the year 2014 [9] Vivekananda Dembre and Jeetendra Gurjar introduced and studied minimal weakly and maximal weakly open maps in topological spaces. Definition 1.1 [1] : A proper non-empty open subset U of a topological space X is said to be minimal open set if any open set which is contained in U is 𝜑 or U. Definition 1.2 [2] : A proper non-empty open subset U of a topological space X is said to be maximal open set if any open set which is contained in U is X or U. Definition 1.3 [3] : A proper non-empty closed subset F of a topological space X is said to be minimal closed set if any closed set which is contained in F is 𝜑 or F. Definition 1.4 [3] : A proper non-empty closed subset F of a topological space X is said to be maximal closed set if any closed set which is contained in F is X or F. Definition 1.5 [4] : Let X and Y be the topological spaces. A bijective function f : X Y is called weakly homeomorphism if both f and f -1 are weakly continuous. Definition 1.6 [5] : Let X and Y be the topological spaces. A bijective function f : X Y is called I) Minimal homeomorphism if both f and f -1 are minimal continuous maps. II) Maximal homeomorphism if both f and f -1 are maximal continuous maps. Definition 1.7 [5] : Let X and Y be the topological spaces
Definition 1.9 [7] : A proper non-empty weakly open subset U of X is said to be minimal weakly open set if any weakly open set which is contained in U is ∅ or U.
Definition 1.10 [8] : A proper non-empty weakly closed subset U of X is said to be maximal weakly open set if any weakly open set which is contained in U is X or U.
Definition 1.11 [9] : Let X and Y be topological spaces.
Minimal regular weakly and maximal regular weakly homeomorphism. Definition 2.1: A bijection function f : X
Y is called I) Minimal regular weakly homeomorphism if both f and f - 1 are minimal regular weaklycontinuous maps. II) Maximal regular weakly homeomorphism if both f and f - 1 are maximal regular weakly continuous maps.
Theorem 2.2:
Every homeomorphism is minimal regular weakly homeomorphism but not conversely.
Proof: Let f : X Y be a homeomorphism. Now f and f -1 are continuous maps then f and f -1 are minimal regular weakly continuous as every continuous map is minimal regular weakly continuous. Hence f is a minimal regular weakly homeomorphism.
Example 2.3: Let X=Y={a,b,c} be with 𝜏 = { X, 𝜑, {a},{a,b}} and 𝜇 = { Y, 𝜑, {a},{b},{a,b} } then f : X Y be a function defined by f(a)=a, f(b)=a and f(c)=c, then f and f -1 are minimal regular weakly continuous maps then f is a minimal regular weakly homeomorphism but it is not a homeomorphism. Since f is not continuous map for the open set {b} in Y ; f -1 ({b}) = b which is not open set in X.
Theorem 2.4: Every homeomorphism is maximal regular weakly homeomorphism but not conversely.
Proof: Let f : X Y be a homeomorphism. Now f and f -1 are continuous maps then f and f -1 are maximal regular weakly continuous as every continuous map is minimal regular weakly continuous. Hence f is a minimal regular weakly homeomorphism.
Example 2.5: Let X = Y = {a,b,c} be with 𝜏 = { X, 𝜑, {a,b}} and 𝜇 = { Y, 𝜑, {a},{b},{a,b} } then f : X Y be a identity function then f and f -1 are maximal regular weakly continuous maps then f is a maximal regular weakly homeomorphism but it is not a homeomorphism. Since f is not a continuous map for the open set {b} in Y ; f -1 ({b}) = b which is not open set in X.
Remark 2.6: Minimal regular weakly homeomorphism and maximal regular weakly homeomorphism are independent of each other.
Example 2.7: In example 2.3 f is minimal regular weakly homeomorphism but it is not maximal regular weakly homeomorphism. In example 2.5 f is maximal regular weakly homeomorphism but it is not minimal regular weakly homeomorphism.
X is a minimal regular weakly homeomorphism and similarly f is minimal regular weakly homeomorphism.
X is a minimal regular weakly homeomorphism and similarly f is minimal regular weakly homeomorphism.
Definition 2.11:
A bijection f : X Y is called (i). min -max regular weakly homeomorphism if both f and f -1 are min -max regular weakly continuous maps. (ii). max-min regular weakly homeomorphism if both f and f -1 are max -min regular weakly continuous maps.
Theorem 2.12: Every min-max regular weakly homeomorphism is minimal regular weakly homeomorphism but not conversely.
Proof : : Let f : X Y be a homeomorphism. Now f and f -1 are continuous maps then f and f -1 are min-max regular weakly continuous as every continuous map is min-max regular weakly continuous. Hence f is a min-max regular weakly homeomorphism.
Example 2.13: In example 2.3 f is a minimal regular weakly homeomorphism but it is not a min-max regular weakly homeomorphism. Since f is not a min-max regular weakly continuous map for the minimal regular weakly open set {b} in Y, f -1 (b) = {b} which is not a maximal regular weakly open set in X.
Theorem 2.14: Every max-min regular weakly homeomorphism is maximal regular weakly homeomorphism but not conversely.
Proof : Let f : X Y be a homeomorphism. Now f and f -1 are continuous maps then f and f -1 are max-min regular weakly continuous as every continuous map is max-min regular weakly continuous. Hence f is a min-max regular weakly homeomorphism. (iii) (i) : Let N be any minimal regular weakly open set in X ; then X-N is a maximal regular weakly closed set in X ; by assumption f (X-N) is a minimal regular weakly closed set in Y. But f (X-N) = (f -1) (X -N) = Y -(f -1) (N) is a minimal regular weakly open set in Y ; therefore (f -1) (N) is a maximal regular weaklyopen set in Y. Hence f -1 :Y X is a minimal-maximal regular weakly homeomorphism and similarly f is min-max regular weakly homeomorphism.
References
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