Homeomorphism between open and closed real intervals

Question

There are 2 topological spaces:

$\mathcal{T}_1$ consists of $\mathbb{R}$, $\emptyset$, and every interval $(−r,r)$, for r any positive real;

$\mathcal{T}_2$ consists of $\mathbb{R}$, $\emptyset$, and every interval $[−r,r]$, and every interval $(−r,r)$, for r any positive real;

Is it possible to build a homeomorphism between $(\mathbb{R}, \mathcal{T}_1)$ and $(\mathbb{R}, \mathcal{T}_2)$ ?

Answer 1

For $\mathcal{T}_2$, there are nonempty open sets which are still open after removing two points (e.g., remove $-1$ and $1$ from $[-1,1]$).

But that is not the case for $\mathcal{T}_1$. Removing two points from a nonempty open set of $\mathcal{T}_1$ yields a set which is not open.

Answer 2

No, the spaces are not homeomorphic. One reason is because $\mathcal T_2$ has non-empty compact open subsets (the closed intervals $[-r,r]$) while $\mathcal T_1$ does not.