Mathematics > Home > Advances in Pure and Applied Mathematics > Issue 1 (January 2025) > Article
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
It is shown that the collection of all topologies on a given set coincide with the set of subsemirings of the power set (equipped with union and intersection) if and only if is finite. Furthermore, given a topological space and a subset of , we characterize when the subspace topology is a maximal (resp., a prime) ideal of the semiring . 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 into in case is a finite set and is an arbitrary nonempty set.
It is shown that the collection of all topologies on a given set coincide with the set of subsemirings of the power set (equipped with union and intersection) if and only if is finite. Furthermore, given a topological space and a subset of , we characterize when the subspace topology is a maximal (resp., a prime) ideal of the semiring . 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 into in case is a finite set and 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