Sudden homotopy: Difference between revisions

Revision as of 17:51, 30 September 2007

This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.

This article defines a property of a homotopy from a topological space to itself

Definition

A sudden homotopy on a topological space $X$ is a continuous map $F : X \times I \to X$ such that $F (x, 0) = x \forall x$ and for any $t < 1$ the map $x \mapsto F (x, t)$ is a homeomorphism from $X$ to $X$ .

Of particular interest is sudden contracting homotopy which is both sudden and contracting.

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 ==Definition==
-A '''sudden homotopy''' on a topological space <math>X</math> is a map <math>F:X \times I \to X</math> such that for any <math>t < 1</math> the map <math>x \mapsto F(x,t)</math> is a homeomorphism from <math>X</math> to <math>X</math>.
+A '''sudden homotopy''' on a topological space <math>X</math> is a continuous map <math>F:X \times I \to X</math> such that <math>F(x,0) = x \ \forall \ x</math> and for any <math>t < 1</math> the map <math>x \mapsto F(x,t)</math> is a homeomorphism from <math>X</math> to <math>X</math>.
 Of particular interest is [[sudden contracting homotopy]] which is both sudden and contracting.