We prove that P-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph _{\omega } $-many random reals to a model of CH.These results improve and correct previous theorems that can be found in the literature.

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ${{\omega }_{1}}$ , as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ${{\omega }_{1}}$ .

In this note we partially answer a question of Cascales, Orihuela and Tkachuk [‘Domination by second countable spaces and Lindelöf ${\rm\Sigma}$-property’, Topology Appl.158(2) (2011), 204–214] by proving that under $CH$ a compact space $X$ is metrisable provided $X^{2}\setminus {\rm\Delta}$ can be covered by a family of compact sets $\{K_{f}:f\in {\it\omega}^{{\it\omega}}\}$ such that $K_{f}\subset K_{h}$ whenever $f\leq h$ coordinatewise.

It is shown that it follows from PFA that there is no compact scattered space of height greater than $\omega $ in which the sequential order and the scattering heights coincide.

If A and B are disjoint ideals on ω, there is a tower preserving σ-centered forcing which introduces a subset of ω which meets every infinite member of A in an infinite set and is almost disjoint fromeverymember of B. We can then produce a model in which all compact separable radial spaces are Fréchet, thus answering a question of P. Nyikos. The question of the existence of compact ccc radial spaces which are not Fréchet was first asked by Chertanov (see [Arh78]).

It is well known that every Boolean algebra of size ω1 can be embedded into (ω)/fin. E. van Douwen proved that if CH failed then there is a Boolean algebra of size ω2 which cannot be embedded into (ω1)/ctble. We show that ◇ is equivalent to the statement that a certain natural Boolean algebra of size embeds into (ω1)/ctble. I would like to thank B. Velickovic for many helpful conversations.

It is shown that the measure algebra on the space 22ω is σ-n-linked for each n ∊ ω.

Although this paper is concerned with answering a question in general topology about sequential convergence, the techniques used may be of greater interest to those interested in the structure of filters on ω. A point in a topological space is called an α1-point if whenever is a countable family of sequences converging to it, there is a sequence B, also converging to it, such that A\B is finite for each A ∈ . A space is α1 if each point of the space is an α1-point. Nyikos has shown that the existence of countable Fréchet α1-spaces which are not first countable follows from the assumption and has asked if it is consistent that no such space exists. It is not difficult to see that if there is an α1-space which is not first countable the there is also one with only one non-isolated point. Since, in answering Nyikos' question, we are only concerned with countable spaces, it follows that we are really dealing with certain types of filters on ω. The precise translation of the topological question to a filter theoretic one is contained in §2. The reader who is not interested in the details may wish to read only §2, the first half of §3 and §5.

0. Introduction. A point p ∈ βX\X is called a remote point of X if P ∉ clβX A for each nowhere dense subset A of X. If X is a topological sum Σ{Xn : n ∈ ω} we call nice if {n : F ∩ Xn = ∅} is finite for each . We call remote if for each nowhere dense subset A of X there is an with F ∩ A = ∅ and n-linked if each intersection of at most n elements of is non-empty.

Absolute C-embeddings have been studied extensively by C. E. Aull. We will use his notation P = C[Q] to mean that a space satisfying property Q is C-embedded in every space having property Q that it is embedded in if (and only if) it has property P. The first result of this type is due to Hewitt [5] where he proves that if Q is “Tychonoff” then P is almost compactness. Aull [2] proves that if Q is “T4 and countable pseudocharacter” or “T4 and first countable” then P is “countably compact”. In this paper we show that P is almost compactness if Q is “Tychonoff” and any of countable pseudocharacter, perfect, or first countability. Unfortunately for the last case we require the assumption that . Finally we show that P is countable compactness if Q is Tychonoff and “closed sets have a countable neighborhood base”. In each of the above results C-embedding may be replaced by C*-embeddings and the results hold if restricted to closed embeddings.