Difference between revisions of "Topological retraction/Definition"
From Maths
m (Typo) |
m (Alec moved page Retraction/Definition to Topological retraction/Definition without leaving a redirect: Retraction is a thing in category theory too) |
(No difference)
| |
Latest revision as of 08:04, 13 December 2016
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be considered a s subspace of [ilmath]X[/ilmath]. A continuous map, [ilmath]r:X\rightarrow A[/ilmath] is called a retraction if[1]:
- The restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath] (the map [ilmath]r\vert_A:A\rightarrow A[/ilmath] given by [ilmath]r\vert_A:a\mapsto r(a)[/ilmath]) is the identity map, [ilmath]\text{Id}_A:A\rightarrow A[/ilmath] given by [ilmath]\text{Id}_A:a\mapsto a[/ilmath]
If there is such a retraction, we say that: [ilmath]A[/ilmath] is a retract[1] of [ilmath]X[/ilmath].
