Difference between revisions of "871 - Topology"
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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>. | 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=== | ||
Revision as of 19:14, 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 \( \mathcal{T}_4 \).
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 \(B\sube V \)
\(T_4 \) (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 \(B \sube V \).
