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

• $X,\varnothing \in \tau$,
• if $𝒰\subseteq \tau$, then $\bigcup _{U\in 𝒰}U\in \tau$,
• if $U_1,U_2,\cdots ,U_n\in \tau$, then $\bigcap _{i=1}^n{U}_i\in \tau$.

Elements of $\tau$ 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 $x\in (a,b)\subseteq U$.