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**Continuous Bijection f:X-->X not a Homeo.**

Hi, All:

A standard example of a continuous bijection that is not a homeomorphism is the

map f:[0,1)-->S^1 : x-->(cosx,sinx) ; for one, S^1 is compact, but [0,1) is not,so

they cannot be homeomorphic to each other.

Now, I wonder if it is possible to do this for a continuous bijection of a space to itself,

(with different topologies if necessary) and, if it is possible from a space with itself ,

but a map g: (X,T)-->(X,T) , i.e., with the same topology for domain and codomain.