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I can find sources that tell me explicitly what an NDR-pair is in terms of a map and a homotopy, but is there a good intuitive idea corresponding to, or some canonical examples that I might be able to investigate?


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up vote 4 down vote accepted

I found this article to be helpful:

Basically, $(X, A)$ is an NDR-pair when there is an open $U \supset A$ (a neighborhood of $A$) such that $A$ is a deformation-retract of $U$. If $A$ is closed in $X$, then this is equivalent to the inclusion $A \hookrightarrow X$ being a cofibration.

I believe the most useful and ubiquitous examples (to an algebraic topologist) are CW-pairs $(X, A)$.

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This isn't quite correct - in an NDR pair the deformation retraction of U onto A is also extended to a continuous homotopy of maps defined on all of X, and there is also a function $u: X \rightarrow [0,1]$ which is zero precisely on A and for which $U = u^{-1}([0,1))$. If you want an intuitive characterization that implies all of this, you can assume that $A \subset X$ has a regular neighborhood, which is a closed neighborhood that is homeomorphic to the mapping cylinder of some map $B \rightarrow A$. – Cary Mar 23 '14 at 2:57

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