Any quick examples of a Normal Space but not $T_1$

Question

So I am having trouble picturing a space that is Normal, but not $T_1$ (if it is, we call it $T_4$). All distinct points in a normal space can be contained in disjoint closed sets that can be separated by disjoint open sets. I am looking for maybe something simple like point sets $\tau_x = \{ \phi, X, \{a,c\},\{a\},\{c\} \}$.

Answer 1

Take a space with the indiscrete topology and more than one point. It is trivially normal since there are no nonempty disjoint closed sets, but it is not $T_1$.

Answer 2

Take a set $X$ and $p \in X$ and define the excluded point topology on $X$

$$\mathcal{T} = \{X\} \cup \{A \subseteq X: p \notin A\}$$

This is $T_0$ but not $T_1$ (as $\{p\}$ is dense, not closed). And all non-empty closed sets, as complements of open sets, contain $p$, so there are no closed disjoint sets to separate at all.