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A bound for the density of any Hausdorff space
We show, in a certain specific sense, that both the density and the
cardinality of a Hausdorff space are related to the "degree" to which the space
is nonregular. It was shown by Sapirovskii that for
a regular space and the author observed this holds if the space is only
quasiregular. We generalize this result to the class of all Hausdorff spaces by
introducing the nonquasiregularity degree , which is countable when
is quasiregular, and showing for any Hausdorff
space . This demonstrates that the degree to which a space is
nonquasiregular has a fundamental and direct connection to its density and,
ultimately, its cardinality. Importantly, if is Hausdorff then is
"small" in the sense that . This results in a unified proof
of both Sapirovskii's density bound for regular spaces and Sun's bound
for the cardinality of a Hausdorff space . A
consequence is an improved bound for the cardinality of a Hausdorff space.Comment: 6 page
Anal fissure
A corrected version of this article is available in the file AnalFissureCorrected.This issue of eMedRef provides information to clinicians on the pathophysiology, diagnosis, and therapeutics of anal fissures
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