Vol. 17, No. 2, 1966

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Metrizability and completeness in normal Moore spaces

Donald Reginald Traylor

Vol. 17 (1966), No. 2, 381–390

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 20 < 21, 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.

Mathematical Subject Classification
Primary: 54.38
Received: 1 November 1964
Published: 1 May 1966
Donald Reginald Traylor