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 $\text{[math]}$ is a topological space on set $\text{[math]}$ , if $\text{[math]}$ satisfies the following properties:

$\text{[math]}$ ,
if $\text{[math]}$ , then $\text{[math]}$ ,
if $\text{[math]}$ , then $\text{[math]}$ .

Elements of $\text{[math]}$ are called open sets.

Examples

The real line $\text{[math]}$ equipped with the standard topology is a topological space. Topology is defined by taking all open intervals as a basis. A set $\text{[math]}$ is open, if for all point $\text{[math]}$ , there exists an open interval $\text{[math]}$ such that $\text{[math]}$ .