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 $τ \subseteq 𝒫 (X)$ is a topology, satisfying 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.
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, the finite intersection property holds.

Therefore, $τ_{ℝ}$ is indeed a topology on $ℝ$ .

□

Discrete topology

Let $X$ be an arbitary set and define the discrete topology of $X$ by $τ = 𝒫 (X)$ . Every subset of $X$ is open in $τ$ .

Proof

$\emptyset, X \subseteq X$ , thus $\emptyset, X \in τ$ .
Let $X \supseteq A, B \in τ$ , since $A$ and $B$ are subsets of $X$ , their union $A \cup B$ and intersection $A \cap B$ are also a subsets of $X$ , which are in the topology $τ$ .

Therefore the discrete topology of $X$ is a topology.

□

Indiscrete topology

Let $X$ be an arbitary set, the indiscrete topology of $X$ is defined by $τ = {\emptyset, X}$ .

Proof

By definition, $\emptyset, X \in τ$ .
$\emptyset \cup X = X \in τ$ .
$\emptyset \cap X = \emptyset \in τ$ .

Therefore the indiscrete topology of $X$ is a topology.

□

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: