# 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. 1.

An identity map is always a bijection.

2. 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. 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

Title identity map IdentityMap 2013-03-22 14:03:43 2013-03-22 14:03:43 bwebste (988) bwebste (988) 7 bwebste (988) Definition msc 03E20 identity mapping identity operator identity function ZeroMap IdentityMatrix