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=== | ||
Revision as of 19:11, 7 January 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 \(\mathcal{T}_4<math>.
==='"`UNIQ--h-2--QINU`"'Connectedness===
==='"`UNIQ--h-3--QINU`"'Path Connected===
==='"`UNIQ--h-4--QINU`"'Compactness===
:''see also:'' [[Compact]]
==='"`UNIQ--h-5--QINU`"'Separation Properties ===
===='"`UNIQ--h-6--QINU`"'<math>T_1 \) ====
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 \(B\sube V \)
\(T_4 \) (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 \( A \sube U\) and \(B \sube V \).
