Zoltán Szentmiklóssy

Cardinal restrictions on some homogeneous compacta

We give restrictions on the cardinality of compact Hausdorff homogeneous spaces that do not use other cardinal invariants, but rather covering and separation properties. In particular, we show that it is consistent that every hereditarily normal homogeneous compactum is of cardinality c. We introduce property wD(κ), intermediate between the properties of being weakly κ-collectionwise Hausdorff and strongly κ-collectionwise Hausdorff, and show that if X is a compact Hausdorff homogeneous space in which every subspace has property wD(N 1 ), then X is countably tight and hence of cardinality < 2 c . As a corollary, it is consistent that such a space X is first countable and hence of cardinality c. A number of related results are shown and open problems presented.

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.

On the cardinality of certain Hausdorff spaces

Abstract We prove (in ZFC) the following theorem. Assume κ is an infinite cardinal, X is a Hausdorff space such that every subspace Y of X is the union of κ compact subsets of Y . Then X has cardinality at most κ.

Discrete subspaces of countably tight compacta

Abstract Our main result is that the following cardinal arithmetic assumption, which is a slight weakening of GCH, “ 2 κ is a finite successor of κ for every cardinal κ ”, implies that in any countably tight compactum X there is a discrete subspace D with | D ¯ | = | X | . This yields a (consistent) confirmation of Alan Dow’s Conjecture 2 from [A. Dow, Closures of discrete sets in compact spaces, Studia Math. Sci. Hung. 42 (2005) 227–234].

What makes a space have large weight

Abstract In Section 2 of this paper we formulate several conditions (two of them are necessary and sufficient) which imply that a space of small character has large weight. In Section 3 we construct a ZFC example of a 0-dimensional space X of size 2 ω with w ( X )=2 ω and χ ( X )= nw ( X )= ω , we show that CH implies the existence of a 0-dimensional space Y of size ω 1 with w ( Y )= nw ( Y )= ω 1 and χ ( Y )= R ( Y )= ω , and we prove that it is consistent that 2 ω is as large as you wish and there is a 0-dimensional space Z of size 2 ω such that w ( Z )= nw ( Z )=2 ω but χ ( Z )= R ( Z ω )= ω .

Pinning down versus density

AbstractThe pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for any neighborhood assignment U: X → τX there is a set A ∈ [X]κ with A ∩ U(x) ≠ Ø for all x ∈ X. Clearly, c(X) ≤ pd(X) ≤ d(X).Here we prove that the following statements are equivalent(1) 2κ < κ+ω for each cardinal κ(2) d(X) = pd(X) for each Hausdorff space X (3) d(X) = pd(X) for each 0-dimensional Hausdorff space X.This answers two questions of Banakh and Ravsky.The dispersion character Δ(X) of a space X is the smallest cardinality of a non-empty open subset of X. We also show that if pd(X) < d(X) then X has an open subspace Y with pd(Y) < d(Y) and |Y| = Δ(Y), moreover the following three statements are equiconsistent(i) There is a singular cardinal λ with pp(λ) > λ+, i.e., Shelah’s Strong Hypothesis fails(ii) there is a 0-dimensional Hausdorff space X such that |X| = Δ(X) is a regular cardinal and pd(X) < d(X)(iii) there is a topological space X such that |X| = Δ(X) is a regular cardinal and pd(X) < d(X).We also prove that• d(X) = pd(X) for any locally compact Hausdorff space X • for every Hausdorff space X we have 

On compact fibered spaces

\left| X \right| \leqslant {2^{{2^{pd\left( X \right)}}}}

The projective -character bounds the order of a -base

and pd(X)<d(X) implies