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 $\text{[math]}$ is a topological space.

The standard topology on $\text{[math]}$ 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 $\text{[math]}$ .

Proof

$\text{[math]}$ vacuously; every point $\text{[math]}$ belongs to some open interval, like $\text{[math]}$ , which is open in $\text{[math]}$ . Therefore by the definition, $\text{[math]}$ .
Let $\text{[math]}$ be open sets and $\text{[math]}$ . Take $\text{[math]}$ , then $\text{[math]}$ for some $\text{[math]}$ . Because $\text{[math]}$ is open, there exists $\text{[math]}$ such that $\text{[math]}$ . Therefore $\text{[math]}$ is open.
The finite intersection property can be proved by induction. Let $\text{[math]}$ be open and $\text{[math]}$ . By definition of openness, there exists $\text{[math]}$ and $\text{[math]}$ such that $\text{[math]}$ . Set $\text{[math]}$ and $\text{[math]}$ . Then $\text{[math]}$ . Thus $\text{[math]}$ is open. By induction,
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Basis

for every $\text{[math]}$ , there exists $\text{[math]}$ with $\text{[math]}$ ,
if $\text{[math]}$ with $\text{[math]}$ , then there exists $\text{[math]}$ such that $\text{[math]}$ .

The topology generated by $\text{[math]}$ consists of all unions of elements of $\text{[math]}$ .

Topological properties

Some key properties of topological spaces include: