What is the difference between isomorphism and homeomorphism? I have some questions understanding isomorphism. Wikipedia said that 

isomorphism is bijective homeomorphism

I kown that $F$ is a homeomorphism if $F$ and $F^{-1}$ are continuous. So my question is: If $F$ and its inverse are continuous, can it not be bijective? Any example? I think if $F$ and its inverse are both continuous, they ought to be bijective, is that right?

Isomorphism and homeomorphism appear both in topology and abstract algebra. Here, I think you mean isomorphism and homeomorphism in topology. In one word, they are the same in topology.
