What are closure and boundary of $(-\infty,\infty)\times (0,T)$

Question

For $T>0$, define $$ \Omega_T:=(-\infty,\infty)\times (0,T). $$

I wonder how the closure $\overline{\Omega}_T$ and the boundary $\partial\Omega_T$ look like.

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For example, if we want to impose boundary conditions I need to know what $\partial\Omega_T$ is... or another scenario: Applying some maximum principle which tells me there cant be a minima in the interior... what is the boundary, what the interior here?

Answer 1

In the usual topology you have $$ \partial \Omega_T=\partial\{(x,y)\in\mathbb R~:~0<y<T\}=\{(x,y)~:~y=0\text{ or }y=T\}=(\mathbb R\times\{0\})\cup (\mathbb R\times\{T\}). $$ And $\Omega_T$ is open, so the interior of $\Omega_T$ is $\Omega_T$ itself.

But this is much more basic than the maximum principle...

Answer 2

The set $\Omega_T$ is a subset of $S = (-\infty,\infty)\times [0,T]$, which is a closed set in $\mathbb R \times \mathbb R$. So the closure of $\Omega_T$ is contained in

$\tag 1 S= \mathbb R \times \{0\} \; \bigcup \; \Omega_T \; \bigcup \; \mathbb R \times \{T\}$

Now every open rectangle containing the $(x, 0)$ must intersect $\Omega_T$, and a similar statement can be made for any point $(x, T) \in \mathbb R \times \{T\}$. So the closure of $\Omega_T$ is indeed $S$. Also, it is easy to see that the $\partial\Omega_T$ is given by

$\tag 2 S= \mathbb R \times \{0\} \; \bigcup \; \mathbb R \times \{T\}$