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