A normal space that is not regular, how is it possible?

Question

I can see on Math Stack that Sierpinski space is normal but not regular. Has is this possible ?

My definitions :

Regular $(X,T)$ is regular if for all $x\in X$ and all closed $A$ there are disjoint open $U$ and $V$ s.t. $x\in U$ and $A\subset V$.

Normal $(X,T)$ is normal if for all closed $A$ and $B$ there are disjoint open $U$ and $V$ s.t. $A\subset U$ and $B\subset V$.

So, how can Normal doesn't implies Regular ? By the way, in the Munkres (page 195) they say that : "it's clear that a normal space is regular"... apparently not that much since Sierpinski in a counter example... I don't understand anything, is Munkres wrong ?

By the way, it's clear that Regular implies Hausdorff, no ? (or Munkres failed an other time ?)

Answer 1

Munkres assumed in regular and normal spaces both that one-point sets are closed (i.e. $X$ is $T_1$) this is only mentioned in the first sentence in the definition on p 195 (2nd edition):

Suppose that one-point sets are closed in $X$. Then $X$ is said to be regular if for each pair consisting of a point $x$ and closed set $B$ disjoint from $x$, there exist disjoint open sets containing $x$ and $B$ respectively. The space $X$ is said to be normal if for each pair $A,B$ of disjoint closed sets of $X$ there exist disjoint open sets containing $A$ and $B$ respectively.

The next sentence after this definition is: "It's clear that regular spaces are Hausdorff, and that a normal space is regular".

This is indeed clear under the one-point set is closed extra condition: we can take singleton sets for the closed set in the regular case and get point separations, and we can take a singleton $A$ to get regular from normal.

Often a regular (in the more general sense you mean: separating points and disjoint closed sets) with closed singletons together is called $T_3$ and normal (separating disjoint closed sets) plus closed singletons is called $T_4$. We then get the chain of implications $T_4 \rightarrow T_3$ (as we saw) and $T_3 \rightarrow T_2$ (where $T_2$ is Hausdorff) and $T_2 \rightarrow T_1$ ($T_1$ is that one-point sets are closed). Munkres doesn't really define $T_1$ or $T_0$ and does not really consider them.

With notions like normal and regular always check the definitions an author gives; usages vary and can be confusing.