Contractibility is not closure-preserved: Difference between revisions

Latest revision as of 15:27, 31 May 2016

This article gives the statement, and possibly proof, of a topological space property (i.e., contractible space) not satisfying a topological space metaproperty (i.e., closure-preserved property of topological spaces).
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Definition

It is possible to have a topological space $X$ and a subset $A$ of $X$ such that $A$ is a contractible space in the subspace topology from $X$ , but $\bar{A}$ , the closure of $A$ in $X$ , is not contractible.

Related facts

Proof

Set $X$ to be a circle and set $A$ to be the complement of a single point in the circle. Then, $A$ is homeomorphic to the real line, and is contractible. However, it closure $\bar{A}$ equals $X$ , which is not even simply connected, and therefore not contractible.

This example can be generalized to the $n$ -sphere $S^{n}$ for $n \geq 1$ , where the complement of any point is homeomorphic to $R^{n}$ (via, for instance, a stereographic projection).

@@ Line 6: / Line 6: @@
 It is possible to have a [[topological space]] <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is a [[contractible space]] in the [[subspace topology]] from <math>X</math>, but <math>\overline{A}</math>, the closure of <math>A</math> in <math>X</math>, is not contractible.
+== Related facts ==
+* [[Contractibility is not interior-preserved]]
+* [[Seifert-van Kampen theorem]]
+* [[Union of two simply connected open subsets with path-connected intersection is simply connected]]
 == Proof ==
@@ Line 11: / Line 17: @@
 Set <math>X</math> to be a [[circle]] and set <math>A</math> to be the complement of a single point in the circle. Then, <math>A</math> is homeomorphic to the real line, and is contractible. However, it closure <math>\overline{A}</math> equals <math>X</math>, which is not even simply connected, and therefore not contractible.
-This example can be generalized to the <math>n</math>-[[sphere]] <math>S^n</math> for <math>n \ge 1</math>.
+This example can be generalized to the <math>n</math>-[[sphere]] <math>S^n</math> for <math>n \ge 1</math>, where the complement of any point is homeomorphic to <math>\R^n</math> (via, for instance, a stereographic projection).