Topological space

A topological space is an ordered pair $(X,\mathcal{T})$ where $X$ is a set and $\mathcal{T}\subseteq \mathcal{P}(X)$ is a topology on $X$, meaning:

• $\varnothing\in\mathcal{T}$ and $X\in\mathcal{T}$,
• if $\{U_i\}_{i\in I}\subseteq\mathcal{T}$ then $\bigcup_{i\in I}U_i\in\mathcal{T}$,
• if $U,V\in\mathcal{T}$ then $U\cap V\in\mathcal{T}$.

Here $\mathcal{P}(X)$ denotes the power set of $X$, and the members of $\mathcal{T}$ are the open sets (whose complements are the closed sets ). Many standard constructions—such as the subspace topology , product topology , and quotient topology —produce new topological spaces from existing ones.

Examples:

• $\mathbb{R}$ with its usual topology (open sets are unions of open intervals).
• Any set $X$ with the discrete topology $\mathcal{T}=\mathcal{P}(X)$.
• Any metric space $(X,d)$, using the topology induced by the metric .