@ARTICLE{TBA, 

TITLE={[FORTHCOMING] Algebraic properties of subspace topologies}, 
  
AUTHOR={Noômen Jarboui
, Bana Al Subaiei, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={}, 

NUMBER={Forthcoming papers<br>}, 
	
YEAR={2024}, 

URL={https://openscience.fr/Algebraic-properties-of-subspace-topologies}, 
	
DOI={TBA}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={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.}}