nLab neighborhood retract

Definition

A topological subspace $A$ is a neighborhood retract of a topological space $X$ if there is a neighborhood $U\supset A$ in $X$ such that $A$ is a retract of $U$.

A metrisable topological space $Y$ is an absolute neighborhood retract if for any embedding $Y\subset Z$ as a closed subspace in a metrisable topological space $Z$, $Y$ is a neighborhood retract of $Z$.

A pair $(X,A)$ where $A$ is a closed subspace of $X$ is an NDR-pair (for ‘Neighbourhood Deformation Retract’) or a closed Borsuk pair if there is a function $u:X\to I=[0,1]$ and a homotopy $H:X\times I\to X$ such that $H(x,0)=x$, for all $x\in X$, $H(a,t)=a$ for all $a\in A$, $H(x,1)\in A$ for all $x\in X$ such that $u(x)\lt 1$ and $u^{-1}(0)\subset A$. (See deformation retract.)

Properties

The canonical inclusion $i:A\hookrightarrow X$ corresponding to any NDR-pair $(X,A)$ is a closed Hurewicz cofibration.

Related concepts

• retract, deformation retract

References

Textbook accounts

• Peter May, Section 6.4 of: A concise course in algebraic topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)