We prove that every countably compact AP-space is Fréchet-Urysohn. It is also established that if is a paracompact space and is AP, then is a Hurewicz space. We show that every scattered space is WAP and give an example of a hereditarily WAP-space which is not an AP-space.
We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the never soft and never countably paracompact numbers. We show that these cardinals must both be equal to under the effective weak diamond principle , answering questions of da Silva S.G., On the presence of countable paracompactness, normality and property in spaces from almost disjoint families,...
We consider the question: when does a Ψ-space satisfy property (a)? We show that if then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).
A space is called connectifiable if it can be densely embedded in a connected Hausdorff space. Let be the following statement: “a perfect -space with no more than clopen subsets is connectifiable if and only if no proper nonempty clopen subset of is feebly compact". In this note we show that neither nor is provable in ZFC.
Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define to be X ∩ M with topology generated by . Suppose is homeomorphic to the irrationals; must ? We have partial results. We also answer a question of Gruenhage by showing that if is homeomorphic to the “Long Cantor Set”, then .
MSC 2010: 54A25, 54A35.
We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions with the property that every prime number that divides also divides , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic...