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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 1 (January 2025)   > Article

Algebraic properties of subspace topologies

Propriétés algébriques des sous-espaces de topologies


Noômen Jarboui
University of Sfax
Tunisia

Bana Al Subaiei
King Faisal University
Saudi Arabia



Published on 20 January 2025   DOI : 10.21494/ISTE.OP.2025.1255

Abstract

Résumé

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It is shown that the collection of all topologies on a given set X coincide with the set of subsemirings of the power set P(X) (equipped with union and intersection) if and only if X is finite. Furthermore, given a topological space (X,T) and a subset A of X, we characterize when the subspace topology TA is a maximal (resp., a prime) ideal of the semiring T. As applications, we provide an algebraic characterization of the one-point compactification of a noncompact, Tychonoff space. Moreover, we describe explicitly the semiring homomorphisms from P(X) into P(Y) in case X is a finite set and Y is an arbitrary nonempty set.

It is shown that the collection of all topologies on a given set X coincide with the set of subsemirings of the power set P(X) (equipped with union and intersection) if and only if X is finite. Furthermore, given a topological space (X,T) and a subset A of X, we characterize when the subspace topology TA is a maximal (resp., a prime) ideal of the semiring T. As applications, we provide an algebraic characterization of the one-point compactification of a noncompact, Tychonoff space. Moreover, we describe explicitly the semiring homomorphisms from P(X) into P(Y) in case X is a finite set and Y is an arbitrary nonempty set.

One-point compactification Completely regular topological space Semiring Maximal ideal

One-point compactification Completely regular topological space Semiring Maximal ideal

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