Compact-calibres of regular and monotonically normal spaces (Q1607836)

Summary: A topological space has calibre \(\omega_1\) (resp., calibre \((\omega_1,\omega)\)) if every point-countable (resp., point-finite) collection of nonempty open sets is countable. It has compact-calibre \(\omega_1\) (resp., compact-calibre \((\omega_1,\omega)\)) 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.