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 di...

 In Proposition 2.6 in (G‎. ‎Gruenhage‎, ‎A‎. ‎Lutzer‎, ‎Baire and Volterra spaces‎, ‎textit{Proc‎. ‎Amer‎. ‎Math‎. ‎Soc.} {128} (2000)‎, ‎no‎. ‎10‎, ‎3115--3124) a condition that‎ ‎every point of $D$ is $G_delta$ in $X$ was overlooked‎. ‎So we‎ ‎proved some conditions by which a Baire space is equivalent to a‎ ‎Volterra space‎. ‎In this note we show that if $X$ is a‎ ‎monotonically normal $T_1...

Moody, P. J. and A. W. Roscoe, Acyclic monotone normality, Topology and its Applications 47 (1992) 53-67. A space X is acyclic monotonically normal if it has a monotonically normal operator M(., .) such that for distinct points x,,, ,x._, in X and x, =x,], n::i M(x,, X\{x,+,}) = (d. It is a property which arises from the study of monotone normality and the condition “chain (F)“. In this paper i...

We answer questions of Bennett, Lutzer, and Matveev by showing that any monotonically compact LOTS is metrizable, and any first-countable Lindelöf GO-space is monotoncically Lindelöf. We also show that any compact monotonically Lindelöf space is first-countable, and is metrizable if scattered, and that separable monotonically compact spaces are metrizable.