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In section 31 of Munkres' Topology text on regular and normal spaces, he often assumes for the theorems and interesting results that regular/normal spaces have closed point sets (i.e. the space is $T_1$). But he also makes it clear that the implications $$ \text{normal} \implies \text{regular} \implies \text{Hausdorff}$$ are only valid when the topological space is $T_1$; because Hausdorff spaces are always $T_1$, but that normal and regular spaces need not be $T_1$ (e.g. a two point set in the trivial topology). That much is clear.

However, when working some of the problems, specifically exercise 1, he makes no assumption that the space is $T_1$, just that it is regular. I spent some time on this without assuming the space was $T_1$, which makes the solution quite simple in that case, but I was adamant that I couldn't assume that. After spending too much time thinking about it I caved and looked at the solution, which assumed that one-point sets were closed ($T_1$).

Am I crazy or is Munkres only having us consider regular/normal spaces that are $T_1$ since those are the most interesting?

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  • $\begingroup$ I haven't meet a single non T1 space (with the exeption of the explamples following the definition of a T1 space) so i guess one can live happily with only T1 spaces... $\endgroup$ Sep 24, 2014 at 19:48

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You’re not crazy: you just didn’t look closely enough at his definition. If you look closely at the definition of regularity at the beginning of Section $31$, you’ll see that the property is defined only for spaces in which singletons are closed, i.e., only for $T_1$ spaces. What he calls regular is what I (and other sensible people :-)) call $T_3$, so that the $T_k$ properties form a real hierarchy. Unfortunately, his definition is not uncommon, though Wikipedia uses the definitions that I prefer while noting the variants. Both usages are common enough that you simply have to learn which one any given author uses.

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