István Juhász

HFD and HFC type spaces, with applications

Abstract The aim of this paper is to give a survey of HFD and HFC type spaces. These are subspaces of Cantor cubes 2 λ with very flexible combinatorial properties that yield many basic examples of S and L spaces as well as numerous other interesting topological spaces.

On Jakovlev spaces

A topological spaceX is called weakly first countable, if for every pointx there is a countable family {Cnx |n ∈ω} such thatx ∈Cn+1x ⊆Cnx and such thatU ⊂X is open iff for eachx ∈U someCnx is contained inU. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N. N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V. I. Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption b=c, and also by forcing.

A tall space with a small bottom

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if • <• = • then there is such a space of height • + with onlymany isolated points. This implies that there is a locally compact scattered space of height !2 with !1 isolated points in ZFC, solving an old problem of the flrst author.

Partitioning the pairs and triples of topological spaces

Abstract We carry out the task given by the title, introduce a combinatorial principle, and use it to prove X↛( top ω+1) 2 ω for all spaces X, X ↛ (Y) 3 ω for all spaces X where Y is any nondiscrete countable space, and related results.

The projective -character bounds the order of a -base

All spaces below are Tychonov. We define the projective π-character pπ Χ (X) of a space X as the supremum of the values π Χ (Y) where Y ranges over all (Tychonov) continuous images of X. Our main result says that every space X has a π-base whose order is ≤ pπ Χ (X); that is, every point in X is contained in at most pπ Χ (X)-many members of the π-base. Since pπ Χ (X) < t(X) for compact X, this is a significant generalization of a celebrated result of Shapirovskii.

On -countably tight spaces

Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality

On the tightness of \({G_\delta}\)-modifications

mathfrak{c}

Strong colorings yield -bounded spaces with discretely untouchable points

if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and

D-FORCED SPACES: A NEW APPROACH TO RESOLVABILITY

sigma

Combinatorial Principles from Adding Cohen Reals

-countably tight compactum has cardinality