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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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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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With the ambiguity that tends to accompany separation axioms, I think it prudent to define my terms first:

A regular space is one in which a closed set and a point not contained in it can be separated by open neighborhoods.

A normal space is one in which disjoint closed sets can be separated by open neighborhoods.

The following examples come from $\pi$-Base, which is a searchable database of spaces from Steen and Seebach's Counterexamples in Topology.

The following spaces are regular but not normal. You can learn more about them by viewing the search result.

$[0, \Omega) \times I^I$

Deleted Tychonoff Corkscrew

Deleted Tychonoff Plank

Dieudonne Plank

Hewitt’s Condensed Corkscrew

Michael’s Product Topology

Niemytzki’s Tangent Disc Topology

Rational Sequence Topology

Sorgenfrey’s Half-Open Square Topology

Thomas’s Corkscrew

Thomas’s Plank

Tychonoff Corkscrew

Uncountable Products of $\mathbb{Z}^+$

The following spaces are normal but not regular. You can learn more about them by viewing the search result.

Countable Excluded Point Topology

Divisor Topology

Either-Or Topology

Finite Excluded Point Topology

Hjalmar Ekdal Topology

Nested Interval Topology

Right Order Topology on $\mathbb{R}$

The Integer Broom

Uncountable Excluded Point Topology

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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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