nLab
neighborhood retract
Context
Topology
topology (point-set topology )

see also algebraic topology , functional analysis and homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space (see also locale )

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

sphere , ball ,

circle , torus , annulus

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Basic homotopy theory

Contents
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 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 Hurewicz cofibration .

Revised on April 3, 2012 17:22:21
by

Urs Schreiber
(82.169.65.155)