Types of topological retractions
From Maths
Revision as of 14:45, 10 May 2016 by Alec (Talk | contribs) (Created page with "{{Stub page|grade=A|msg=Break it down into sections, or use a table, whatever, just clean up the layout}} ==Definitions== ===Retraction=== {{:Retraction/Definition}} ===...")
Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Break it down into sections, or use a table, whatever, just clean up the layout
Contents
Definitions
Retraction
Deformation retraction
A subspace, [ilmath]A[/ilmath], of a topological space [ilmath](X,\mathcal{ J })[/ilmath] is called a deformation retract of [ilmath]X[/ilmath], if there exists a retraction[1][2], [ilmath]r:X\rightarrow A[/ilmath], with the additional property:
- [ilmath]i_A\circ r\simeq\text{Id}_X[/ilmath][1][2] (That [ilmath]i_A\circ r[/ilmath] and [ilmath]\text{Id}_X[/ilmath] are homotopic maps)
- Here [ilmath]i_A:A\hookrightarrow X[/ilmath] is the inclusion map and [ilmath]\text{Id}_X[/ilmath] the identity map of [ilmath]X[/ilmath].
Recall that a retraction, [ilmath]r:X\rightarrow A[/ilmath] is simply a continuous map where [ilmath]r\vert_A=\text{Id}_A[/ilmath] (the restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath]). This is equivalent to the requirement: [ilmath]r\circ i_A=\text{Id}_A[/ilmath].
- Caution:Be sure to see the warnings on terminology
Strong deformation retraction
Strong deformation retraction/Definition
