In [4], J. Ceder proved that every paracompact strongly complete semimetrizable space is completely metrizable. This result cannot be generalized to paracompact weakly complete semimetrizable spaces as a known example of L. F. McAuley shows (see [11, Theorem 3.2]). It then arises, in a natural way, the question of obtaining conditions for the complete metrizability of a paracompact weakly complete semimetrizable space. In this note we give an answer to this question. We show that every regular theta, weakly complete asymmetrizable space is a complete Aronszajn space and deduce, among other results, that every paracompact theta, weakly complete semimetrizable space is completely metrizable. Some examples related to the obtained results are also given.
@article{urn:eudml:doc:38463,
title = {Weakly complete semimetrizable spaces and complete metrizability.},
journal = {Extracta Mathematicae},
volume = {11},
year = {1996},
pages = {276-281},
zbl = {0881.54034},
mrnumber = {MR1437451},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38463}
}
Romaguera, Salvador; Shore, Sam D. Weakly complete semimetrizable spaces and complete metrizability.. Extracta Mathematicae, Tome 11 (1996) pp. 276-281. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38463/