homotopy equivalence
Definition Suppose that $X$ and $Y$ are topological spaces^{} and $f:X\to Y$ is a continuous map. If there exists a continuous map $g:Y\to X$ such that $f\circ g\simeq i{d}_{Y}$ (i.e. $f\circ g$ is http://planetmath.org/node/1584homotopic^{} to the identity mapping on $Y$), and $g\circ f\simeq i{d}_{X}$, then $f$ is a homotopy equivalence^{}. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence.
If there exist a homotopy equivalence between the topological spaces $X$ and $Y$, we say that $X$ and $Y$ are homotopy equivalent, or that $X$ and $Y$ are of the same homotopy type. We then write $X\simeq Y$.
0.0.1 Properties

1.
Any homeomorphism $f:X\to Y$ is obviously a homotopy equivalence with $g={f}^{1}$.

2.
For topological spaces, homotopy equivalence is an equivalence relation^{}.

3.
A topological space $X$ is (by definition) contractible^{}, if $X$ is homotopy equivalent to a point, i.e., $X\simeq \{{x}_{0}\}$.
References
 1 A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also available http://www.math.cornell.edu/ hatcher/AT/ATpage.htmlonline.
Title  homotopy equivalence 
Canonical name  HomotopyEquivalence 
Date of creation  20130322 12:13:22 
Last modified on  20130322 12:13:22 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  14 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 55P10 
Related topic  HomotopyOfMaps 
Related topic  WeakHomotopyEquivalence 
Related topic  Contractible 
Related topic  HomotopyInvariance 
Related topic  ChainHomotopyEquivalence 
Related topic  PathConnectnessAsAHomotopyInvariant 
Related topic  TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT 
Defines  homotopy equivalent 
Defines  homotopically equivalent 
Defines  homotopy type 
Defines  strong homotopy equivalence 