identity map

DefinitionIf $X$ is a set, then the identity map in $X$ is the mappingthat 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 ${\tau }_2$. Thenthe 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, ishomotopic (http://planetmath.org/HomotopyOfMaps)to the antipodal map $A:S^n\to S^n$ if $n$ is odd [1].