I am trying to solve the following exercise.
Study the countability axioms, separability axioms and the Lindelöf property in $(X,\tau)$, where $X=[-1,1]$ and $U\in\tau$ if and only if $0\notin U$ or $(-1,1)\subset U$.
Is this the excluded point topology in $[-1,1]$, excluding $0$? I think it is, since the only open set including $0$ is $X$. But I don't know if this is the right conclusion.
If it is, I know that the excluded point topology es second-countable, $T_0$ and Lindelöf (because it is second-countable).
Is this right or I am leaving something?