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B. Fitzpatrick, Jr. and D. R.
Traylor proved [Theorem 1, Pac. J. Math., to appear] that if there is a
normal, nonmetrizable Moore space then there is one which is not locally
metrizable at any point. The primary purpose of this paper is to extend
the stated result to include normal, complete Moore spaces. That is, it is
established that there is a normal, complete, Moore space which is not locally
metrizable at any point, provided there exists a normal, complete, nonmetrizable
Moore space. Indeed, it is further established that, provided there exists a
nonmetrizable, normal, complete Moore space, then there is one which is also
connected, locally connected, not locally metrizable at any point, and, using a
result of Younglove’s [Theorem 1, “Concerning metric subspaces of nonmetric
spaces,” Fund. Math., 48 (1949), 15–25], which contains a dense metrizable
subset.
F. B. Jones [Bull. Amer. Math. Soc. 43 (1937), 671–677] showed that if
2ℵ0 < 2ℵ1, then every normal separable Moore space is metrizable. It is
established in this paper that if each normal, separable, connected space
satisfying Axioms 0,1, and 2 of [R. L. Moore, Foundations of Point Set Theory,
Amer. Math. Soc. Colloq. Pub. No. 13 Providence, R. I. 1962] is metrizable, then
each normal separable Moore space is metrizable.
Other theorems of this ilk are included in this paper.
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