Strong deformation retract: Difference between revisions

Revision as of 22:05, 26 September 2007

This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces

Definition

Symbol-free definition

A subspace of a topological space is termed a deformation retract if there is a homotopy between the identity map on the whole space, and a retraction onto the subspace, such that the map at every intermediate stage, restricts to identity on the subspace.

Definition with symbols

A subspace $A$ of a topolofical space $X$ is termed a deformation retract of $X$ if there is a homotopy $F : X \times I \to X$ such that:

$f (x, 0) = x \forall x \in X$
$f (a, t) = a \forall a \in A, t \in I$
$f (x, 1) \in A \forall x \in X$

The second condition is what distinguishes deformation retracts from the weaker notion of homotopy retract.

Relation with other properties

Weaker properties

@@ Line 2: / Line 2: @@
 ==Definition==
+===Symbol-free definition===
 A subspace of a topological space is termed a '''deformation retract''' if there is a homotopy between the identity map on the whole space, and a retraction onto the subspace, such that the map at every intermediate stage, restricts to identity on the subspace.
+===Definition with symbols===
+A subspace <math>A</math> of a topolofical space <math>X</math> is termed a '''deformation retract''' of <math>X</math> if there is a homotopy <math>F: X \times I \to X</math> such that:
+* <math>f(x,0) = x \forall x \in X</math>
+* <math>f(a,t) = a \forall a \in A, t \in I</math>
+* <math>f(x,1) \in A \forall x \in X</math>
+The second condition is what distinguishes deformation retracts from the weaker notion of homotopy retract.
 ==Relation with other properties==