Topological space - Knowlpedia

A topological space is a pair $(X,\mathcal T)$ , where $X$ is a set and $\mathcal T$ is a collection of subsets of $X$ , satisfying:

Empty set and whole space: $\varnothing,X\in\mathcal T$ .
Arbitrary unions: the union of any collection of members of $\mathcal T$ belongs to $\mathcal T$ .
Finite intersections: the intersection of finitely many members of $\mathcal T$ belongs to $\mathcal T$ .

The collection $\mathcal T$ is the topology of the space.

Constructions

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