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

(2) ( existsM. 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 allM. 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) existsM. M^\infty(X) is metrizable.
(ii) for allM. M^\infty(X) is metrizable.
(iii) (Under (b = \omega₁)), for allM. M(X) is metrizable.
(iv) X is \sigma-discrete and metrizable.

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

(7) ( existsM. 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, ( existsM. 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

Date received: March 5, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caam-48.