Consider [0,1], [0,1), and (0,1) as subspaces of R with the standard topology. Prove that none of these spaces are homeomorphic.
Hint: try removing points and see what happens with cutsets.
Based on your hint, suppose we have a homeomorphism between [0,1] and [0,1) such that
.
If h is homeomorphism, then the restriction of h removing 1 from the domain of h and h(1) from the codomain should be homeomorphism such that
.
We see that is not a homeomorphism because the domain of is connected but the codomain of is not connected. Contradiction.
Thus, h is not a homeomorphism either.
The remaining cases are similar to the above one.