The separation of the subset $[-1, 0) \cup (0,1)$ in $\mathbb{R}$ contains the limit point of the other

Question

In the book of Topology by Munkres, it is given at page 149 that

However, what does the author mean by the fact that $0$ is the common limit point of the two open sets in $Y$ does not matter ?, and how can this be ? I mean isn't this a contradiction to lemma 23.1 ?

Answer 1

$A=[-1,0)$, $0$ is a limit point of $A$, but $B=(0,1]$ does not contain the limit point: $0\notin B$.

Similarly, $0$ is a limit point of $B$, but $A$ does not contain it.

No contradiction is involved.