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Almost 2-Fully Normal, Pairwise Paracompact and Complete Developable Bispaces
Abstract
We introduce and study the notion of an almost 2-fully normal
bispace. In particular; we prove that a bispace is quasi-pseudometrizable if
and only if it is almost 2-fully normal and pairwise developable. We obtain
conditions under which an almost 2-fully normal bispace is subquasi-metrizable
and show that the fine quasi-uniformity of any subquasi-metrizable topological
space is bicomplete. We prove that every pairwise paracompact bispace (in the
sense of Romguera and Marin, 1988) is almost 2-fully normal and that the finest
quasi-uniformity of any 2-Hausdorff pairwise paracompact bispace is bicomplete.
We also characterize pairwise paracompactness in terms of a property of $\sigma$-Lebesgue
type of the finest quasi-uniformity. Finally, we use Salbany's compactification
of pairwise Tychonoff bispaces to characterize those bispaces that admit a
bicomplete pair development and deduce that an interesting example of R. Fox
of a non-quasi-metrizable pairwise stratifiable pairwise developable bispace
admits a bicomplete pair development.
Mathematics Subject Classification (1991): 54E55, 54E35, 54E15, 54E30
Keywords: bispace, almost 2-fully normal, pair development, quasi-metrizable,
subquasi-metrizable, finest quasi-uniformity, bicomplete, pairwise paracompact,
s-Lebesgue, Salbany's compactification, bitopologies, metric spaces, metrizability,
uniform structures and generalizations, Moore spaces, study, quasi-uniformity,
topological space, compactification, Tychonoff
Quaestiones Mathematicae
24(1) 2001, 21–37
bispace. In particular; we prove that a bispace is quasi-pseudometrizable if
and only if it is almost 2-fully normal and pairwise developable. We obtain
conditions under which an almost 2-fully normal bispace is subquasi-metrizable
and show that the fine quasi-uniformity of any subquasi-metrizable topological
space is bicomplete. We prove that every pairwise paracompact bispace (in the
sense of Romguera and Marin, 1988) is almost 2-fully normal and that the finest
quasi-uniformity of any 2-Hausdorff pairwise paracompact bispace is bicomplete.
We also characterize pairwise paracompactness in terms of a property of $\sigma$-Lebesgue
type of the finest quasi-uniformity. Finally, we use Salbany's compactification
of pairwise Tychonoff bispaces to characterize those bispaces that admit a
bicomplete pair development and deduce that an interesting example of R. Fox
of a non-quasi-metrizable pairwise stratifiable pairwise developable bispace
admits a bicomplete pair development.
Mathematics Subject Classification (1991): 54E55, 54E35, 54E15, 54E30
Keywords: bispace, almost 2-fully normal, pair development, quasi-metrizable,
subquasi-metrizable, finest quasi-uniformity, bicomplete, pairwise paracompact,
s-Lebesgue, Salbany's compactification, bitopologies, metric spaces, metrizability,
uniform structures and generalizations, Moore spaces, study, quasi-uniformity,
topological space, compactification, Tychonoff
Quaestiones Mathematicae
24(1) 2001, 21–37
