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identity map
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(Definition)
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Definition If $X$ is a set, then the identity map in $X$ is the mapping that maps each element in $X$ to itself.
- An identity map is always a bijection.
- 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., $\tau_1\subset\tau_2$ .
- The identity map on the $n$ -sphere, is homotopic to the antipodal map $A:S^n\to S^n$ if $n$ is odd [1].
- 1
- V. Guillemin, A. Pollack, Differential topology, Prentice-Hall Inc., 1974.
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"identity map" is owned by bwebste. [ full author list (2) | owner history (1) ]
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Cross-references: odd, antipodal map, finer, continuous, topologies, bijection, mapping
There are 123 references to this entry.
This is version 4 of identity map, born on 2003-11-01, modified 2006-10-15.
Object id is 5418, canonical name is IdentityMap.
Accessed 12748 times total.
Classification:
| AMS MSC: | 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory ) |
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Pending Errata and Addenda
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