International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 4, Pages 209-216
doi:10.1155/S0161171202011365
Compact-calibres of regular and monotonically normal spaces
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Received 10 January 2001; Revised 10 May 2001
Copyright © 2002 David W. Mcintyre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A topological space has calibre ω1 (resp., calibre (ω1,ω)) if every point-countable (resp.,
point-finite) collection of nonempty open sets is countable. It
has compact-calibre ω1 (resp., compact-calibre (ω1,ω)) if, for every family of uncountably many nonempty open sets, there is some compact set which meets uncountably many (resp., infinitely many) of them. It has CCC (resp., DCCC) if every disjoint (resp., discrete) collection of
nonempty open sets is countable. The relative strengths of these six conditions are determined for Moore spaces, regular first countable spaces, linearly-ordered spaces, and arbitrary regular spaces. It is shown that the relative strengths for spaces with point-countable bases are the same as those for Moore spaces, and the relative strengths for linearly-ordered spaces are the same
as those for arbitrary monotonically normal spaces.