identity map
Definition If $X$ is a set, then the identity map in $X$ is the mapping that maps each element in $X$ to itself.
0.0.1 Properties

1.
An identity map is always a bijection.

2.
Suppose $X$ has two topologies^{} ${\tau}_{1}$ and ${\tau}_{2}$. Then the identity mapping $I:(X,{\tau}_{1})\to (X,{\tau}_{2})$ is continuous if and only if ${\tau}_{1}$ is finer than ${\tau}_{2}$, i.e., ${\tau}_{1}\subset {\tau}_{2}$.

3.
The identity map on the $n$sphere, is homotopic (http://planetmath.org/HomotopyOfMaps) to the antipodal map $A:{S}^{n}\to {S}^{n}$ if $n$ is odd [1].
References
 1 V. Guillemin, A. Pollack, Differential topology, PrenticeHall Inc., 1974.
Title  identity map 

Canonical name  IdentityMap 
Date of creation  20130322 14:03:43 
Last modified on  20130322 14:03:43 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  7 
Author  bwebste (988) 
Entry type  Definition 
Classification  msc 03E20 
Synonym  identity mapping 
Synonym  identity operator 
Synonym  identity function 
Related topic  ZeroMap 
Related topic  IdentityMatrix 