Petr Simon

On minimal π-character of points in extremally disconnected compact spaces

Abstract We consider the relationship between π-character, refinement number (=weak density) and π-weight in complete Boolean algebras. As an application we shall show that every extremally disconnected compact space contains a point which is not an accumulation point of any countable discrete subset, provided that minimal π-character and π-weight coincide.

Spaces which are generated by discrete sets

Abstract Continuing the study initiated by Dow, Tkachenko, Tkachuk and Wilson, we prove that countably compact countably tight spaces, normed linear spaces in the weak topology and function spaces over σ-compact spaces are discretely generated. We also construct, using [CH], a compact pseudoradial space and a pseudocompact space of countable tightness which are not discretely generated.

On the Pytkeev property in spaces of continuous functions

Answering a question of Sakai, we show that the minimal cardi- nality of a set of reals X such that Cp(X) does not have the Pytkeev property is equal to the pseudo-intersection number p. Our approach leads to a natu- ral characterization of the Pytkeev property of Cp(X) by means of a covering property of X, and to a similar result for the Reznichenko property of Cp(X).

Reaping number and p-character of Boolean algebras

For every Boolean algebra B the minimal π-character of an ultrafilter on B is at most 2r2, where r2 is the reaping number of B. An example of B is given for which r2(B)< min{πχ(U): U ϵ Ult(B)}.

An α4 , not Fréchet product of α4 Fréchet spaces

Abstract We shall construct in ZFC two Frechet–Urysohn α 4 -spaces, the product of which is α 4 , but fails to be Frechet–Urysohn. This answers Noguras question from 1985.

Origins of Dimension Theory

It is well-known that there are three dimension functions recognized today in general topology.

Ultrafilters on ω and atoms in the lattice of uniformities II

Abstract Interaction between ultrafilters and uniformities on a countable set is investigated. Various ultrafilters are constructed such that atoms in the lattice of uniformities refining the corresponding ultrafilter uniformities have special properties.

A countable dense-in-itself dense P-set☆

Abstract We shall show that several rather familiar countable topological spaces are embedded as P-sets in their Cech–Stone compactifications.

On spaces in which every bounded subset is Hausdorff

Abstract The class of spaces in the title (denoted by Haus(e-comp )) is introduced and it is compared with other classes of weak Hausdorff spaces. An explicit description of the Haus(e-comp )- epimorphisms is given by means of a variation of Arhangelskii–Franklins compactly determined closure. It is shown that Haus(e-comp ) is not co-well-powered.

Completely separable MAD families

Publisher Summary This chapter provides an overview of completely separable maximal almost disjoint (MAD) families. An infinite family A ⊆ [ w ] w is almost disjoint if any two of its distinct elements have finite intersection. A family A is said to be a MAD family if it is almost disjoint and for every X ∈ [ w ] w there is an A ∈ A such that A ∩ X is infinite. There are almost disjoint (hence also MAD) families of cardinality c and many MAD families with special combinatorial and/or topological properties can be constructed using set-theoretic assumptions like CH, MA or b = c . However, special MAD families are notoriously difficult to construct in ZFC alone. The reason being the lack of a device ensuring that a recursive construction of a MAD family would not prematurely terminate, an object that would serve a similar purpose as independent linked families do for the construction of special ultra-filters. The notion of a completely separable MAD family is a candidate for such a device and is an interesting notion in its own right.