A topological space is said to be -separable if has a -closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that -separable PIGO spaces are perfect and asked if -separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of -separable monotonically normal spaces which are not perfect. Extremely normal -separable spaces are shown to be stratifiable.
Hušek defines a space X to have a small diagonal if each uncountable subset of X² disjoint from the diagonal has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω₁ which has a small diagonal will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable; in particular, Juhász and Szentmiklóssy proved...
Strongly sequential spaces were introduced and studied to solve a problem of Tanaka concerning the product of sequential topologies. In this paper, further properties of strongly sequential spaces are investigated.
It is proved that the product of two pseudo radial compact spaces is pseudo radial provided that one of them is monolithic.
A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique fixed point...
We study closed subspaces of -Ohio complete spaces and, for uncountable cardinal, we prove a characterization for them. We then investigate the behaviour of products of -Ohio complete spaces. We prove that, if the cardinal is endowed with either the order or the discrete topology, the space is not -Ohio complete. As a consequence, we show that, if is less than the first weakly inaccessible cardinal, then neither the space , nor the space is -Ohio complete.
We consider the question of when , where is the elementary submodel topology on X ∩ M, especially in the case when is compact.
We construct in ZFC an ultrafilter such that for every one-to-one function there exists with in the summable ideal, i.e. the sum of reciprocals of its elements converges. This strengthens Gryzlov’s result concerning the existence of -points.