---
type: "concept"
title: "Topological space"
slug: "topological-space"
language: "en"
translationGroupId: "cmr24dsv1000bn101plljexi9"
domain: "General topology"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

A topological space is a [[Set|set]] $X$ together with a [[topology|topology]] $\tau$, that is, a collection of subsets of $X$, called the [[Open set|open sets]], satisfying the following axioms:
1. The [[Empty set|empty set]] and the whole space are open: $\emptyset\in \tau$ and $X\in \tau$.
2. [[Arbitrary union|Arbitrary unions]] of open sets are open: if $\{U_{i}\}_{i\in I}$ is any family of sets in $\tau$, then $\bigcup_{i \in I} U_i \in \tau$.
3. Finite [[Intersection of sets|intersections]] of open sets are open: if $U_1, \ldots, U_n \in \tau$, then $U_1 \cap \cdots \cap U_n \in \tau$.