Topological space: Difference between revisions

Revision as of 14:26, 12 April 2026

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 $\text{[math]}$ .

□

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.

□

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.

□

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:

@@ Line 2: / Line 2: @@
 == Definition ==
-An [[ordered pair]] <math>(X, \tau)</math> is a topological space on set <math>X</math>, if <math>\tau\subseteq \mathcal{P}(X)</math> satisfies the following properties:
+An [[ordered pair]] <math>(X, \tau)</math> is a topological space on set <math>X</math>, if <math>\tau\subseteq \mathcal{P}(X)</math> is a '''topology''', satisfying the following properties:
 * <math>X, \varnothing \in \tau</math>,
@@ Line 11: / Line 11: @@
 == Examples ==
-The real line <math>\mathbb{R}</math> equipped with the standard topology <math>\tau_{\mathbb{R}}</math> is a topological space.
+=== Standard topology ===
+The real line <math>\mathbb{R}</math> equipped with the '''standard topology''' <math>\tau_{\mathbb{R}}</math> is a topological space.
 The standard topology on <math>\mathbb{R}</math> is defined by taking all open intervals as a [[basis]]. A set <math>U\subseteq \mathbb{R}</math> is open, if for all point <math>x\in U</math>, there exists an open interval <math>(a,b)</math> such that <math>x\in (a,b)\subseteq U</math>.
@@ Line 17: / Line 19: @@
 * <math>\varnothing\in\tau_{\mathbb{R}}</math> vacuously; every point <math>x\in\mathbb{R}</math> belongs to some open interval, like <math>(x-1,x+1)</math>, which is open in <math>\mathbb{R}</math>. Therefore by the definition, <math>\mathbb{R}\in\tau_{\mathbb{R}}</math>.
 * Let <math>\{U_i\}_{i\in I}</math> be open sets and <math>U=\bigcup_{i\in I}U_i</math>. Take <math>x\in U</math>, then <math>x\in U_i</math> for some <math>i\in I</math>. Because <math>U_i</math> is open, there exists <math>(a,b)</math> such that <math>x\in (a,b)\subset U_i\subset U</math>. Therefore <math>U</math> is open.
-* The finite intersection property can be proved by induction. Let <math>U, V</math> be open and <math>x\in U\cap V</math>. By definition of openness, there exists <math>(a_1,b_1)\subset U</math> and <math>(a_2,b_2)\subset V</math> such that <math>(a_1,b_1)\ni x \in (a_2,b_2)</math>. Set <math>a=\max\{a_1,a_2\}</math> and <math>b=\min\{b_1,b_2\}</math>. Then <math>x\in(a,b)\subset (a_1,b_1)\cap(a_2,b_2)\subset U\cap V</math>. Thus <math>U\cap V</math> is open. By induction, <math></math>
+* Let <math>U, V</math> be open and <math>x\in U\cap V</math>. By definition of openness, there exists <math>(a_1,b_1)\subset U</math> and <math>(a_2,b_2)\subset V</math> such that <math>(a_1,b_1)\ni x \in (a_2,b_2)</math>. Set <math>a=\max\{a_1,a_2\}</math> and <math>b=\min\{b_1,b_2\}</math>. Then <math>x\in(a,b)\subset (a_1,b_1)\cap(a_2,b_2)\subset U\cap V</math>. Thus <math>U\cap V</math> is open. By induction, the finite intersection property holds.
+Therefore, <math>\tau_\mathbb{R}</math> is indeed a topology on <math>\mathbb{R}</math>.
 }}
+=== Discrete topology ===
+Let <math>X</math> be an arbitary set and define the '''discrete topology''' of <math>X</math> by <math>\tau=\mathcal{P}(X)</math>. Every subset of <math>X</math> is open in <math>\tau</math>.
+{{Proof|proof=* <math>\emptyset, X\subseteq X</math>, thus <math>\emptyset, X\in \tau</math>.
+* Let <math>X\supseteq A,B\in \tau</math>, since <math>A</math> and <math>B</math> are subsets of <math>X</math>, their union <math>A\cup B</math> and intersection <math>A\cap B</math> are also a subsets of <math>X</math>, which are in the topology <math>\tau</math>.
+Therefore the discrete topology of <math>X</math> is a topology.}}
+=== Indiscrete topology ===
+Let <math>X</math> be an arbitary set, the '''indiscrete topology''' of <math>X</math> is defined by <math>\tau=\{\emptyset, X\}</math>.
+{{Proof|proof=* By definition, <math>\emptyset, X\in \tau</math>.
+* <math>\emptyset \cup X=X\in\tau</math>.
+* <math>\emptyset \cap X=\emptyset\in\tau</math>.
+Therefore the indiscrete topology of <math>X</math> is a topology.}}
 ==Basis==