TITLE: Show that no two of the spaces $(0,1), \ (0, 1], \ [0,1]$ are homeomorphic.
QUESTION [7 upvotes]: I want to show that no two of the spaces $(0,1), \ (0, 1], \ [0,1]$ are homeomorphic, with the hint: what happens if you remove a point from each one of these spaces.
No idea where to begin, except that it concludes a chapter on connectedness.
All these spaces are all in the subspace topology of the usual topology on $\mathbb{R}$.
Edit
(Thanks @Chris Culter)
So far what I've got is:
Let $A = (0,1] - \{1\}$.  If $f:(0,1] \rightarrow (0,1)$ is a homeomorphism, then $f$ restricted to $A$ is a homeorphism onto a subset of $(0,1)$ with one and only one point removed.  But any removal of a point in $(0,1)$ results in a disconnected space, while at the same time $f_A$ ensures that the space is connected, it being a continuous image of a connected space $A$.  Therefore there is no continuous bijective map between the two spaces let alone a homeomorphism.  A similar agument shows that $[0,1]$ and $(0,1)$ are not homeomorphic.  
I can't seem to show that $(0,1]$ and $[0,1]$ are not homeomorphic using this method though.

REPLY [6 votes]: Well, following the hint: if you remove a point from $(0,1)$, is the remaining space connected?

REPLY [6 votes]: All of them have a different number of non-cut-points, i.e. points that you can remove without disconnecting the set.