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].