Zoltan Balogh

A normal screenable nonparacompact space in ZFC

We construct a normal, screenable, nonparacompact space in ZFC. The existence of such a space is also known to imply that there is a normal, screenable space which is not collectionwise normal.

Reflecting point-countable families

It is shown that if every ≤ ω 1 -sized subspace of a (regular) space X of density ≤ ω 1 has a point-countable base, then so does X. Similar results hold for meta-Lindelofness. Dows reflection theorem and a number of other results are deduced as corollaries and applications.

On collectionwise normality of locally compact, normal spaces

We prove that by adjoining supercompact many Cohen or random reals to a model of ZFC set theory, in the resulting model, every normal locally compact space is collectionwise normal. In the same models, countably paracompact, locally compact T 3 -spaces are expandable. Local compactness in the above theorems can be weakened to being of point-countable type, a condition that is implied by both Cech-completeness and first countability

Nonshrinking open covers and K. Morita's Third Conjecture

Abstract For every regular cardinal κ we construct a hereditarily normal, countably paracompact space X κ which has an increasing open cover W = { W μ : μ ϵ ω 1 } such that W has no refinement by ⩽ κ closed sets. By work of Chiba, Przymusinski and Rudin, the existence of such a space proves a conjecture of Morita that σ-locally compact metrizable spaces are precisely the spaces whose product with every normal, countably paracompact space is normal. The spaces X κ are the first known examples of hereditarily normal ω 1 -Dowker spaces.

When the collection of ε-balls is locally finite

Consider the class N of metrizable spaces which admit a metric d such that, for every e>0, the collection {B(x,ɛ):x∈X} of all e-balls is locally finite. We show that N is precisely the class of strongly metrizable spaces, i.e., X∈N iff X is homeomorphic to a subspace of κω×[0,1]ω for some cardinal κ (where κ carries the discrete topology). In particular, this shows that not every metrizable space admits such a metric, thereby answering a question of Nagata.

Base multiplicity in compact and generalized compact spaces

Abstract We show that a compact Hausdorff space is metrizable if it has a base B such that every countably infinite subset of X is contained in at most countably many members of B . We show that the same statement for countably compact spaces is consistent with and independent of ZFC . These results answer questions stated by Arhangelskii et al. [Topology Appl. 100 (2000) 39–46]. We prove some strenthenings of these theorems. We also consider generalizations of our results to higher cardinalities as well as to wider classes of spaces.

There is a

It is shown that there is a regular T 1 -space whose every subset isi a G δ -set and yet the space is not σ-discrete

Q

We construct a paracompact space QX such that every subset of QX is an Fσ-set, yet QX is not σ-discrete. We will construct our space not to have a Gδ-diagonal, which answers questions of A.V. Arhangel ′skǐı and D. Shakhmatov on cleavable spaces.

-set space in ZFC

Abstract A topological property P is called n -additive in n th power (or weakly n -additive) if a topological space X has P as soon as X n = ∪{ Y i : i ϵ n } where all Y i have P. If P is n -additive in n th power for all natural n ⩾ 1, we say that P is weakly finitely additive. The main question we deal with in this paper is whether metrizability is weakly finitely additive. It was proved by Tkachuk (1994) that it is so in the class of regular spaces with Souslin property. Metrizability was also proved by Tkachuk (1994) to be weakly finitely additive in the class of Hausdorff compact spaces. We generalize this last result, showing that metrizability is weakly finitely additive in the class of regular pseudocompact spaces. We also prove that if X n is a regular Lindelof space then it is metrizable if represented as a union of its n metrizable subspaces. We show that there is an example of a Tychonoff nonmetrizable space X such that X n is a union of two metrizable subspaces for all n ⩾ 1. The method of constructing this example can be used to solve several problems stated by Tkachuk (1994).

There is a paracompact Q-set space in ZFC

It is shown that Con(ZF) implies Con(ZFC + there exists a Dowker filter on ω 2 ). The problem is also considered on higher cardinals.