The family of open intervals that do not contain $0$

Question

Let $T$ be the collection of all open sets in $\mathbb{R}$ not containing $0$ union $\mathbb{R}$ i.e $$T=\{(a,b)\subset\mathbb{\bar R}:0\notin(a,b)\}\cup\{\mathbb{R}\}$$

Then what is true about $T$?

$1.$Hausdorff

$2.$Compact

$3.$Connected

My try:It is not Hausdorff because for $0\neq1$ we can not find two disjoint open sets $U,V$ containing separately $0,1$.

About other two options I do not have any idea.

Thanks.

Answer 1

I'm assuming that you mean $T$ is the smallest topology containing these sets - $T$ itself is not for example closed under unions.

Here is a thought for 2: How many open sets in your topology include the point $0$? What might this imply about an open cover in this topology?