I have 3 spaces A, B, C.

A is compact, the other 2 are not.

A and B are NOT homeomorphic, neither are A & C.

Suppose I can construct a continuous bijection a: C-->B & another b:B-->A.

How can I use this information to decide whether a is a homeomorphism?

Or is this NOT enough info?

If that is the case, I can tell you a bit more about the specific spaces I'm dealing with...

Many thanks in advance x