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 $\text{[math]}$ is a topology, satisfying 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

Standard topology

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.
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, the finite intersection property holds.

Therefore, $\text{[math]}$ is indeed a topology on $\text{[math]}$ .

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Discrete topology

Let $\text{[math]}$ be an arbitary set and define the discrete topology of $\text{[math]}$ by $\text{[math]}$ . Every subset of $\text{[math]}$ is open in $\text{[math]}$ .

Proof

$\text{[math]}$ , thus $\text{[math]}$ .
Let $\text{[math]}$ , since $\text{[math]}$ and $\text{[math]}$ are subsets of $\text{[math]}$ , their union $\text{[math]}$ and intersection $\text{[math]}$ are also a subsets of $\text{[math]}$ , which are in the topology $\text{[math]}$ .

Therefore the discrete topology of $\text{[math]}$ is a topology.

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Indiscrete topology

Let $\text{[math]}$ be an arbitary set, the indiscrete topology of $\text{[math]}$ is defined by $\text{[math]}$ .

Proof

By definition, $\text{[math]}$ .
$\text{[math]}$ .
$\text{[math]}$ .

Therefore the indiscrete topology of $\text{[math]}$ is a topology.

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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: