A Topologic Space $(\mathbb R$,$\tau)$ is VACOUSLY normal but not regular

Question

$\tau$ $= \{ ]-\infty,a [$ ; $a$ $\in \mathbb R$ }

Show that $(\mathbb R$,$\tau)$ is VACOUSLY normal but not regular.

We know in this topologic space closed sets are the form $[b,\infty]$ $(\exists b \in \mathbb R)$

However, I couldn't find any disjoint subset. Is there any mistake? If someone would help me it will be very very beneficial. I am totally confused. Thanks

Answer 1

The statement of normality is "for all pairs $A,B$ of disjoint non-empty closed subsets $A$ and $B$ have disjoint open neighbourhoods". As there are no pairs of disjoint non-empty closed sets, this statement is automatically true. Or seen in another light: To show that $X$ is not normal, we need to produce two disjoint non-empty closed sets that do not have disjoint open neighbourhoods, and we cannot as no such pairs exist. So we cannot refute it, so it is true (excluded middle).

We can refute regularity: $0 \notin [1,\infty)$ and the latter set is closed. But in $X$ all non-empty open sets intersect, so this point and this closed set not containing the point (there are plenty of those pairs to find) do not have disjoint open neighbourhoods as well.