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Is it true that, in a topological space $(X, \mathcal{T})$, regularity does not imply normality and vice versa?

I looked for examples to prove this; but I just don't know many examples to look into. If it is really true, can anyone give one or two good examples for both the cases? Regards.

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2 Answers

up vote 2 down vote accepted

I presume you're working with the definitions that say that a space is regular/normal if a point and a closed set/two closed sets can be separated by open neighbourhoods.

An example of normality not implying regularity is the excluded point topology. For concreteness, set $X=\mathbb{N}$ and let the open sets be $\mathbb{N}$ and any subset not containing 0. Then all nonempty closed sets contain 0, so the space is normal vacuously, but it cannot be regular, since any open neighbourhood of a closed set must in fact be the whole space.

Note, however, that normality implies regularity if your space is $T_1$.

A regular space which isn't normal is slightly trickier to cook up. The following is a slightly simplified presentation of what is usually called the deleted Tychonoff plank. Let $A$ be a discrete countable space, $B$ a discrete uncountable space and $A^*,B^*$ their one-point compactifications. Let $X=(A^*\times B^*)\setminus\{(\infty_A,\infty_B)\}$. Since both $A$ and $B$ are locally compact and Hausdorff, $A^*$ and $B^*$ are Hausdorff, so $A^*\times B^*$ is compact and Hausdorff. Since $X$ is an open subspace of the product, it is locally compact and Hausdorff. Hence, by a standard result, it is regular.

To see that $X$ isn't normal, you can try to prove that the sets $M=\{(\infty_A,b);b\in B\}$ and $N=\{(a,\infty_B);a\in A\}$ are disjoint closed sets with no disjoint open neighbourhoods.

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The following examples come from Spacebook, which is a searchable database of spaces from Steen and Seebach's Counterexamples in Topology. More information about them can be found in the text or on the web.

These examples are regular but not normal. (Sadly, the database contains none that are normal, but not regular. Other answers provide examples, however.)

Deleted Tychonoff Corkscrew

Deleted Tychonoff Plank

Dieudonne Plank

Hewitt’s Condensed Corkscrew

Michael’s Product Topology

Niemytzki’s Tangent Disc Topology

Open Uncountable Ordinal Crossed with Uncountable Cartesian Product of Unit Interval

Rational Sequence Topology

Sorgenfrey’s Half-Open Square Topology

Thomas’s Corkscrew

Thomas’s Plank

Tychonoff Corkscrew

Uncountable Products of Positive Integers

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It's possible that there are two different definitions of normal floating around here. The one I use requires that the space be $T_1$ and two closed sets can be separated by open sets containing each. (Under this definition, therefore, normality implies regularity.) –  Austin Mohr Dec 11 '12 at 4:03
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Indeed; the Steen & Seebach book is notable for using the convention that regular and normal imply $T_1$, in opposition to what the general trend seems to have become. –  Miha Habič Dec 11 '12 at 4:06
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