Show that no two of the spaces $(0, 1), (0, 1], [0, 1]$ are homeomorphic 
Show that no two of the spaces $(0, 1),  (0, 1], [0, 1]$ are homeomorphic

My Attempted Proof
We have $(0, 1) \subset (0, 1] \subset [0, 1]$.
We also have $(0, 1] = (0, 1) \cup \{1\}$ and thus $|(0, 1)| \neq |(0, 1]|$ and thus no bijective mapping $f : (0, 1) \to (0, 1]$ exists, hence $(0,1)$ and $(0, 1]$ are not homeomorphic regardless of the topology defined on them.
The proof for the other cases are similar. $\ \square$

Is my proof correct? I've seen proofs which form a contradiction through connectedness. Is there any error in my arguments?

Use the commonly used method of removing points from your space. Removing any point from $ (0, 1) $ disconnects it, while you can remove at most $ 1 $ point from $ (0, 1] $ without disconnecting it, and at most $ 2 $ from $ [0, 1] $. Since homeomorphism preserves connected components, the result follows.
