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

Standard topology

The real line $ℝ$ equipped with the standard topology $τ_{ℝ}$ is a topological space.

The standard topology on $ℝ$ 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 $x \in (a, b) \subseteq U$ .

Proof

$\emptyset \in τ_{ℝ}$ vacuously; every point $x \in ℝ$ belongs to some open interval, like $(x - 1, x + 1)$ , which is open in $ℝ$ . Therefore by the definition, $ℝ \in τ_{ℝ}$ .
Let ${U_{i}}_{i \in I}$ be open sets and $U = ⋃_{i \in I} U_{i}$ . Take $x \in U$ , then $x \in U_{i}$ for some $i \in I$ . Because $U_{i}$ is open, there exists $(a, b)$ such that $x \in (a, b) \subset U_{i} \subset U$ . Therefore $U$ is open.
The finite intersection property can be proved by induction. Let $U, V$ be open and $x \in U \cap V$ . By definition of openness, there exists $(a_{1}, b_{1}) \subset U$ and $(a_{2}, b_{2}) \subset V$ such that $(a_{1}, b_{1}) ∋ x \in (a_{2}, b_{2})$ . Set $a = \max {a_{1}, a_{2}}$ and $b = \min {b_{1}, b_{2}}$ . Then $x \in (a, b) \subset (a_{1}, b_{1}) \cap (a_{2}, b_{2}) \subset U \cap V$ . Thus $U \cap V$ is open. By induction,

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Basis

for every $x \in X$ , there exists $B \in ℬ$ with $x \in B$ ,
if $x \in B_{1} \cap B_{2}$ with $B_{1}, B_{2} \in ℬ$ , then there exists $B_{3} \in ℬ$ such that $x \in B_{3} \subseteq B_{1} \cap B_{2}$ .

The topology generated by $ℬ$ consists of all unions of elements of $ℬ$ .

Topological properties

Some key properties of topological spaces include: