Topological space: Difference between revisions

Revision as of 14:25, 25 March 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 $\text{[math]}$ 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:

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 == Examples ==
-The real line <math>\mathbb{R}</math> equipped with the standard topology is a topological space. Topology 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>.
+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>.
+{{Proof|proof=
+* <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>
+}}
+==Basis==
+* for every <math>x \in X</math>, there exists <math>B \in \mathcal{B}</math> with <math>x \in B</math>,
+* if <math>x \in B_1 \cap B_2</math> with <math>B_1, B_2 \in \mathcal{B}</math>, then there exists <math>B_3 \in \mathcal{B}</math> such that <math>x \in B_3 \subseteq B_1 \cap B_2</math>.
+The topology generated by <math>\mathcal{B}</math> consists of all unions of elements of <math>\mathcal{B}</math>.
+== Topological properties ==
+Some key properties of topological spaces include:
+* [[Compact space|Compactness]]
+* [[Connected space|Connectedness]]
+* [[Hausdorff space|Hausdorff property]]
+* [[Second-countable space|Second countability]]