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
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1.
An identity map is always a bijection.
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2.
Suppose X has two topologies
τ1 and τ2. Then the identity mapping I:(X,τ1)→(X,τ2) is continuous if and only if τ1 is finer than τ2, i.e., τ1⊂τ2.
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3.
The identity map on the n-sphere, is homotopic (http://planetmath.org/HomotopyOfMaps) to the antipodal map A:Sn→Sn if n is odd [1].
References
- 1 V. Guillemin, A. Pollack, Differential topology, Prentice-Hall Inc., 1974.
Title | identity map |
---|---|
Canonical name | IdentityMap |
Date of creation | 2013-03-22 14:03:43 |
Last modified on | 2013-03-22 14:03:43 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 7 |
Author | bwebste (988) |
Entry type | Definition |
Classification | msc 03E20 |
Synonym | identity mapping |
Synonym | identity operator |
Synonym | identity function |
Related topic | ZeroMap |
Related topic | IdentityMatrix |