Algebraic properties of subspace topologies

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.

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.

One-point compactification Completely regular topological space Semiring Maximal ideal

One-point compactification Completely regular topological space Semiring Maximal ideal