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
$\displaystyle h:[0,1] \rightarrow [0,1)$.
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
$\displaystyle \bar{h}:[0,1]\setminus\{1\} \rightarrow [0,1)\setminus\{h(1)\}$.
We see that $\displaystyle \bar{h}$ is not a homeomorphism because the domain of $\displaystyle \bar{h}$ is connected but the codomain of $\displaystyle \bar{h}$ is not connected. Contradiction.
Thus, h is not a homeomorphism either.
The remaining cases are similar to the above one.