The 1997 Spring Topology and Dynamics Conference April 10-12, 1997 Lafayette, Louisiana

Normality and metrizability in derived Moore spaces by G.M. Reed (Oxford University )

In [On chain conditions in Moore spaces, General Topology and Its Applications 4 (1974), 255-267], the author gave constructions M(X) and M^\infty(X) which produce Moore spaces for any regular, first countable T₃ -space, where M represents a particular choice of countable local bases for each element of X . These constructions have since often been used by the author and others to answer open questions in the literature. In this paper the author considers conditions on X under which the two constructions are normal or metrizable. The results are surprising and provide answers to open questions on the productivity of normality in Moore spaces.

For example, it is shown that it is consistent with ZFC that there exists a normal, separable, locally compact Moore space X such that X² is not normal. Recall that (1) (V=L) implies that normal locally compact Moore spaces are metrizable and (2) (MA) implies that normal, separable, locally compact Moore spaces have normal squares. It is also shown that it is consistent with ZFC and GCH that there exists a normal Moore space with a nonnormal square.

Examples of general theorems (where X is a regular, first countable T₃ -space) include the following:

(1) (there exists a M . M(X) is normal) implies that each subset of X is a G_\delta -set.

(2) (there exists a M . M(X) is metrizable) implies that X is a \sigma -discrete Moore space with a \sigma -disjoint base.

(3) There exists a \sigma -discrete Moore space X of cardinality \omega₁ with a \sigma -disjoint base such that (for every M . M(X) is not normal).

(4) There exists a Moore space with M₁ such that M₁(X) is metrizable and M₂ such that M₂(X) is not normal, and it is consistent with ZFC that such a space exists which is also collectionwise Hausdorff.

(5) The following are equivalent:

• [(i)] there exists a M . M^\infty(X) is metrizable.
• [(ii)] for every M . M^\infty(X) is metrizable.
• [(iii)] (Under (b = \omega₁) ), for every M . M(X) is metrizable.
• [(iv)] X is \sigma -discrete and metrizable.

(6) (b > \omega₁ ) There exists a nonnormal X such that (for every M . M(X) is metrizable).

(7) (there exists a M . M^\infty(X) is normal) implies that both M(X) and X are normal.

(8) If X is metrizable and each subset of X is a G_\delta -set, there exists M such that M^\infty(X) is normal.

(9) (|X| < b ) If X is normal, the following are equivalent:

• [(i)] Each subset of X is a G_\delta -set.
• [(ii)] M(X) is normal.

(10) Under (V=L), if X is normal, (there exists a M . M(X) is normal) implies that X is \sigma -discrete and metrizable.

(11) It is consistent with ZFC that there exists a (non-\sigma -discrete) metrizable space X with each subset of X a G_\delta -set such that:

• [(i)] There exists M₁ such both M₁(X) and M₁^\infty(X) are normal.
• [(ii)] There exists M₂ such that M₂(X) is normal but M₂^\infty(X) is not normal.
• [(iii)] There exists M₃ such both M₃(X) and M₃^\infty(X) are not normal