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# 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 to the antipodal map $A:S^{n}\to S^{n}$ if $n$ is odd [1].

# References

- 1
V. Guillemin, A. Pollack,
*Differential topology*, Prentice-Hall Inc., 1974.

Related:

ZeroMap, IdentityMatrix

Synonym:

identity mapping, identity operator, identity function

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03E20*no label found*

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