Strong deformation retract: Difference between revisions

Latest revision as of 19:59, 11 May 2008

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 strong deformation retract (sometimes simply 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. Such a homotopy is termed a strong deformation retraction.

Definition with symbols

A subspace $A$ of a topological space $X$ is termed a strong deformation retract (sometimes simply 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

Metaproperties

Transitivity

This property of subspaces of topological spaces is transitive. In other words, if $A$ satisfies the property as a subspace of $B$ and $B$ satisfies the property as a subspace of $C$ then $A$ satisfies the property as a subspace of $C$

If $A$ is a strong deformation retract of $B$ and $B$ is a strong deformation retract of $C$ then $A$ is a deformation retract of $C$ .

Template:DP-closed subspace property

If $A_{i}$ is a deformation retract of $B_{i}$ for $i=1,2$ then $A_{1}\times A_{2}$ is a deformation retract of $B_{1}\times B_{2}$ .

@@ Line 5: / Line 5: @@
 ===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.
+A subspace of a topological space is termed a '''strong deformation retract''' (sometimes simply 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. Such a homotopy is termed a [[strong deformation retraction]].
 ===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:
+A subspace <math>A</math> of a topological space <math>X</math> is termed a '''strong deformation retract''' (sometimes simply '''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(x,0) = x \ \forall \ x \in X</math>
-* <math>f(a,t) = a \forall a \in A, t \in I</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>
+* <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]].
-The second condition is what distinguishes deformation retracts from the weaker notion of homotopy retract.
 ==Relation with other properties==
@@ Line 22: / Line 23: @@
 * [[Homotopy retract]]
 * [[Retract]]
+* [[Neighbourhood retract]]
+==Metaproperties==
+{{transitive subspace property}}
+If <math>A</math> is a strong deformation retract of <math>B</math> and <math>B</math> is a strong deformation retract of <math>C</math> then <math>A</math> is a deformation retract of <math>C</math>.
+{{DP-closed subspace property}}
+If <math>A_i</math> is a deformation retract of <math>B_i</math> for <math>i=1,2</math> then <math>A_1 \times A_2</math> is a deformation retract of <math>B_1 \times B_2</math>.