Topological Space

In the setting of Sets and Maps: Ordinary Notation, let $X$ be a set and let $\mathcal{T}$ be a subset of $\mathcal{P}(X)$.

(i) $\mathcal{T}$ is a topology on $X$ if it satisfies the three conditions empty-whole, unions and intersections below.

(ii) $\emptyset\in\mathcal{T}$ and $X\in\mathcal{T}$.

(iii) For every subset $\mathcal{S}$ of $\mathcal{T}$, $\bigcup\mathcal{S}\in\mathcal{T}$.

(iv) For all $U,V\in\mathcal{T}$, $U\cap V\in\mathcal{T}$.

(v) A topological space is a pair $(X,\mathcal{T})$ of a set $X$ and a topology $\mathcal{T}$ on $X$.

(vi) The elements of $\mathcal{T}$ are the open sets of $(X,\mathcal{T})$, and a subset $U$ of $X$ is open in $(X,\mathcal{T})$ if $U\in\mathcal{T}$.