identity map (original) (raw)

0.0.1 Properties

1. 1. An identity map is always a bijection.
2. 1. Suppose X has two topologies τ1 and τ2. Then the identity mapping I:(X,τ1)→(X,τ2) is continuous if and only ifτ1 is finer than τ2, i.e., τ1⊂τ2.
3. 1. 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.