Titre : [FORTHCOMING] Algebraic properties of subspace topologies 

Auteurs : Noômen Jarboui
, Bana Al Subaiei,  

Revue : Advances in Pure and Applied Mathematics 

Numéro : Forthcoming papers<br> 

Volume :  

Date : 2024/07/8 

DOI : TBA 

ISSN : 1869-6090 

Résumé : It is shown that the collection of all topologies on a given set $$$X$$$ coincide with the set of subsemirings of the power set $$$\mathcal{P}(X)$$$ (equipped with union and intersection) if and only if $$$X$$$ is finite. Furthermore, given a topological space $$$(X, \mathcal{T})$$$ and a subset $$$A$$$ of $$$X$$$, we characterize when the subspace topology $$$\mathcal{T}_A$$$ is a maximal (resp., a prime) ideal of the semiring $$$\mathcal{T}$$$. 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 $$$\mathcal{P}(X)$$$ into $$$\mathcal{P}(Y)$$$ in case $$$X$$$ is a finite set and $$$Y$$$ is an arbitrary nonempty set. 

Éditeur : ISTE OpenScience