Topological space

A topological space is a fundamental mathematical structure that generalizes the concept of geometrical spaces and continuity. A topological space is equipped with a collection of open sets, capturing the intuitive idea of "nearness" without necessarily defining a metric. Topological spaces are the objects of study in general topology.

Definition

An ordered pair $(X, τ)$ is a topological space on set $X$ , if $τ \subseteq 𝒫 (X)$ satisfies the following properties:

$X, \emptyset \in τ$ ,
if $𝒰 \subseteq τ$ , then $⋃_{U \in 𝒰} U \in τ$ ,
if $U_{1}, U_{2}, \dots, U_{n} \in τ$ , then $⋂_{i = 1}^{n} U_{i} \in τ$ .

Elements of $τ$ are called open sets.

Examples

The real line $ℝ$ equipped with the standard topology is a topological space. Topology is defined by taking all open intervals as a basis. A set $U \subseteq ℝ$ is open, if for all point $x \in U$ , there exists an open interval $(a, b)$ such that $x \in (a, b) \subseteq U$ .