nLab
delayed homotopy
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

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

topological vector space , Banach space , Hilbert space

topological group

topological manifold

cell complex , CW-complex

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
Theorems

Contents
Idea
A homotopy between continuous function s between topological space s is called delayed if it starts out being constant near one boundary of the interval.

(If it is constant near both boundaries we say it has sitting instant s).

Definition
For $I = [0,1]$ the unit interval and $X$ and $Y$ any topological space s, a continuous map $F: X\times I\to Y$ is a delayed homotopy (between $F(-,0)$ and $F(-1))$ if there exist $t_0\gt 0$ such that $F(x,t)=F(x,0)$ for all $0\leq t\leq t_0$ .

Properties
In Dold-fibrations
Delayed homotopies appear in an alternative characterization of Dold fibration s. See there for details.

Smoothing of delayed homotopies
If a continuous homotopy between two smooth function s is delayed at both ends of the inerval it may be approximated by a smooth homotopy . See Steenrod-Wockel approximation theorem .

Revised on October 25, 2010 17:16:55
by

Urs Schreiber
(131.211.232.186)