Some questions about connectedness

Question

Probably somebody have already been asked it before but I am interested not in the solution of the problem but in some moments which are not clear to me. I am going to ask you the following question:

Show that $(0,1)$ and $(0,1]$ are not homeomorphic.

My approach: Suppose that $(0,1)\approx (0,1]$ then $(0,1]\approx(0,1) $ then $(0,1]-\{1\}\approx (0,1)-\{c\}$ where $c=f(1)$ and $f$ is a homeomorphism.

Then $(0,1)\approx(0,c)\cup (c,1)$. But we know that $(0,1)$ is connected but $(0,c)\cup (c,1)$ not connected because homeomorphism preserves connectedness. This is a contradiction.

As we see the solution of this problem is not difficult at all. However some questions bother me:

1) We consider the spaces $(0,1]$ and $(0,1)$ with subspace topology induced from $\mathbb{R}$, right?

2) When we deleting points from $(0,1]$ and $(0,1)$ we get the spaces $(0,1)$ and $(0,c)\cup (c,1)$. In what topologies we consider the last spaces? In subspace topology induced from $\mathbb{R}$ or induced from $(0,1]$ and $(0,1)$, respectively?

3) I know that $(0,1)$ is connected in subspace topology induced from $\mathbb{R}$. But is it still connected in subspace topology induced from $(0,1]$?

Maybe these questions sound a stupid. But as I know the notion of connectedness depends on topology.

And I would be very grateful if anyone can give comments and answers to my questions!

Answer 1

1. Right.
2. It doesn't matter. It will be the same topologies on both cases. If $X$ is a topological space, if $Y\subset X$, if $Z\subset Y$, then, if you consider on $Y$ the subspace topology, the topolgy on $Z$ seen as a subspace of $Y$ and the topology on $Z$ seen as a subspace of $X$ are the same topologies.
3. Since the topolgies of $(0,1)$ as a subspace of $\mathbb R$ and as a subspace of $(0,1]$ are the same, if it is connected with respect to one of them, it is also connected with respect to the other one.