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].

单词	IdentityMap
释义	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]. References 1 V. Guillemin, A. Pollack,Differential topology, Prentice-Hall Inc., 1974.
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