PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
identity map (Definition)

Definition If $ X$ is a set, then the identity map in $ X$ is the mapping that maps each element in $ X$ to itself.

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.



"identity map" is owned by bwebste. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: zero map, identity matrix

Other names:  identity mapping, identity operator, identity function
Log in to rate this entry.
(view current ratings)

Cross-references: odd, antipodal map, finer, continuous, topologies, bijection, maps, mapping
There are 102 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 9300 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)