Topological space - Revision history

InfernalAtom683 at 06:26, 12 April 2026

2026-04-12T06:26:23Z

← Older revision		Revision as of 14:26, 12 April 2026
Line 2:		Line 2:

	== Definition ==		== 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> ~~is a '''topology''', satisfying~~ 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> satisfies the following properties:

	* <math>X, \varnothing \in \tau</math>,		* <math>X, \varnothing \in \tau</math>,
Line 13:		Line 13:

	=== Standard topology ===		=== Standard topology ===
	The real line <math>\mathbb{R}</math> equipped with the ~~'''~~standard topology~~'''~~ <math>\tau_{\mathbb{R}}</math> is a topological space.		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>.		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 19:		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>.		* <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.		* 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.
	* 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.~~		* 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>

	~~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==		==Basis==

InfernalAtom683 at 06:26, 12 April 2026

2026-04-12T06:26:19Z

← Older revision		Revision as of 14:26, 12 April 2026
Line 2:		Line 2:

	== Definition ==		== 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>,		* <math>X, \varnothing \in \tau</math>,
Line 11:		Line 11:

	== Examples ==		== 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>.		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>.		* <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.		* 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==		==Basis==

InfernalAtom683 at 06:25, 25 March 2026

2026-03-25T06:25:39Z

← Older revision		Revision as of 14:25, 25 March 2026
Line 11:		Line 11:

	== Examples ==		== 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]]

InfernalAtom683: Created page with "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 $(X, \tau)$ is a topological space on set $X$ , if
2026-03-22T14:53:22Z

Created page with "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 <math>(X, \tau)</math> is a topological space on set <math>X</math>, if <math..."

New page
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]] <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:

* <math>X, \varnothing \in \tau</math>,
* if <math>\mathcal{U}\subseteq \tau</math>, then <math>\bigcup_{U\in \mathcal{U}}U \in \tau</math>,
* if <math>U_1, U_2, \cdots, U_n\in \tau</math>, then <math>\bigcap_{i=1}^n U_i \in \tau</math>.

Elements of <math>\tau</math> are called [[Open set|open sets]].

== 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>.