nLab
neighborhood retract

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

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 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.)

Properties

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

Revised on April 3, 2012 17:22:21 by Urs Schreiber (82.169.65.155)