Difference between revisions of "871 - Topology"
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===Metrizability=== | ===Metrizability=== | ||
| + | :''see also'' [[Metrizability]] | ||
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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>. | ||
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| + | 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=== | ||
| Line 10: | Line 15: | ||
===Compactness=== | ===Compactness=== | ||
| + | :''see also:'' [[Compact]] | ||
===Separation Properties === | ===Separation Properties === | ||
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====<math>T_3 </math> (Regular) ==== | ====<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\ | + | 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)==== | ====<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 \ | + | 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== | ==Homotopy== | ||
==Fundamental Groups== | ==Fundamental Groups== | ||
Latest revision as of 22:02, 21 May 2023
Contents
Homeomorphism Invariants
Metrizability
- see also Metrizability
A topological space \( (X,\mathcal{T}_X) \) is metrizable if there is a metric \(d \) on \( X\) such that \(\mathcal{T}_X = \mathcal{T}_d \), where \( \mathcal{T}_d \) is the metric topology on \( X\) induced by \( d\).
Metrizability is a homeomorphism invariant. Metrizability is not preserved by quotients, continuous images, or continuous preimages. Metrizable spaces are \( \mathcal{T}_4 \).
Connectedness
Path Connected
Compactness
- see also: Compact
Separation Properties
\(T_1 \)
A topological space \(X \) is \(T_1 \) if for any two distinct points \( a,b\in X\) there are open sets \( U,V\) in \(X \) with \(a\in U, b\not \in U, a\not\in V, b\in V \).
\(T_2 \) (Hausdorff)
- See also T2 (Hausdorff)
\(T_3 \) (Regular)
A topological space \(X \) is \(T_3 \) if it is \(T_1 \) and for any point \( a \in X\) and closed set \(B \in X \) with \(a \not \in B \), there are disjoint open sets \(U,V \in X \) with \(a\in U \) and \(B\subseteq V \)
\(T_4 \) (Normal)
A topological space \(X \) is \(T_4 \) if it is \(T_1 \)and for any two disjoint closed sets \(A,B \in X \) there are disjoint open sets \(U,V \in X \) with \( A \subseteq U\) and \(B \subseteq V \).
