nLab neighborhood retract




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A topological subspace AA is a neighborhood retract of a topological space XX if there is a neighborhood UAU\supset A in XX such that AA is a retract of UU.

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

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


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


Textbook accounts

Last revised on October 19, 2023 at 01:23:11. See the history of this page for a list of all contributions to it.