Difference between revisions of "871 - Topology"
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====<math>T_2 </math> (Hausdorff) ==== | ====<math>T_2 </math> (Hausdorff) ==== | ||
| + | {{See also|T_2 </math> (Hausdorff)}} | ||
====<math>T_3 </math> (Regular) ==== | ====<math>T_3 </math> (Regular) ==== | ||
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====<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 \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 \sube U</math> and <math>B \sube V </math>. | ||
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| + | ==Homotopy== | ||
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| + | ==Fundamental Groups== | ||
Revision as of 18:45, 7 December 2022
Contents
Homeomorphism Invariants
Metrizability
Connectedness
Path Connected
Compactness
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)
\(T_3 \) (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 \(B \sube V \).
