871 - Topology - Revision history

Kfagerstrom: /* Separation Properties */

2023-05-21T22:02:29Z

Separation Properties

2023-01-07T19:14:04Z

Metrizability

2023-01-07T19:11:03Z

Metrizability

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Homeomorphism Invariants

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Compactness

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T_2 (Hausdorff)

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T_2 (Hausdorff)

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T_2 (Hausdorff)

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T_2 (Hausdorff)

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 ====<math>T_3 </math> (Regular) ====
-A topological space <math>X </math> is <math>T_3 </math> if it is <math>T_1 </math> and for any point <math> a \in X</math> and closed set <math>B \in X </math> with <math>a \not \in B </math>, there are disjoint open sets <math>U,V \in X </math> with <math>a\in U </math> and <math>B\sube V  </math>
+A topological space <math>X </math> is <math>T_3 </math> if it is <math>T_1 </math> and for any point <math> a \in X</math> and closed set <math>B \in X </math> with <math>a \not \in B </math>, there are disjoint open sets <math>U,V \in X </math> with <math>a\in U </math> and <math>B\subseteq V  </math>
 ====<math>T_4 </math> (Normal)====
-A topological space <math>X </math> is <math>T_4 </math> if it is <math>T_1 </math>and for any two disjoint closed sets <math>A,B \in X </math> there are disjoint open sets <math>U,V \in X </math> with <math> A \sube U</math> and <math>B \sube V </math>.
+A topological space <math>X </math> is <math>T_4 </math> if it is <math>T_1 </math>and for any two disjoint closed sets <math>A,B \in X </math> there are disjoint open sets <math>U,V \in X </math> with <math> A \subseteq U</math> and <math>B \subseteq V </math>.
 ==Homotopy==
 ==Fundamental Groups==

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 A topological space <math> (X,\mathcal{T}_X) </math> is metrizable if there is a metric <math>d </math> on <math> X</math> such that <math>\mathcal{T}_X = \mathcal{T}_d </math>, where <math> \mathcal{T}_d </math> is the metric topology on <math> X</math> induced by <math> d</math>.
- Metrizability is a homeomorphism invariant. Metrizability is not preserved by quotients, continuous images, or continuous preimages. Metrizable spaces are <math>\mathcal{T}_4<math>.
+Metrizability is a homeomorphism invariant. Metrizability is not preserved by quotients, continuous images, or continuous preimages. Metrizable spaces are <math> \mathcal{T}_4 </math>.
 ===Connectedness===

← Older revision		Revision as of 19:11, 7 January 2023
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	===Metrizability===		===Metrizability===
		+	:''see also'' [[Metrizability]]
		+
		+	A topological space <math> (X,\mathcal{T}_X) </math> is metrizable if there is a metric <math>d </math> on <math> X</math> such that <math>\mathcal{T}_X = \mathcal{T}_d </math>, where <math> \mathcal{T}_d </math> is the metric topology on <math> X</math> induced by <math> d</math>.
		+
		+	Metrizability is a homeomorphism invariant. Metrizability is not preserved by quotients, continuous images, or continuous preimages. Metrizable spaces are <math>\mathcal{T}_4<math>.

	===Connectedness===		===Connectedness===

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 ==Homeomorphism Invariants==
-[[#Connectedness]]
 ===Metrizability===

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 ==Homeomorphism Invariants==
+[[#Connectedness]]
 ===Metrizability===

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 ====<math>T_2 </math> (Hausdorff) ====
-:''See also'' [[T<sub>2</sub> (Hausdorff)]]
+:''See also'' [[ T2 (Hausdorff)]]
 ====<math>T_3 </math> (Regular) ====