Cardinal restrictions on some homogeneous compacta

Abstract

Abstract. 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(ℵ 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.

Cardinality restrictions on power homogeneous T5 compacta

“…The following notion was used in [6], the terminology we introduce here is mentioned before Problem 3.12 in [6].…”

“…The following lemma is proved in [6]. Since we do not allow cardinal functions to be finite, we exclude the possibility of isolated points in the statement of the following result.…”

“…Partially answering another question of J. van Mill, it was shown by I. Juhász, P. Nyikos and Z. Szentmiklóssy in [6] that consistently every homogeneous T 5 compactum is of cardinality ≤ c. In [15], R. de la Vega proved that every homogeneous compact space of countable tightness is of cardinality ≤ c. Applying the results in [6] and a result of F. D. Tall from [14], it follows from the bound proved by de la Vega that consistently every homogeneous T 5 compactum is first countable. The bound proved by de la Vega for homogeneous compacta was also proved for power homogeneous compacta by A. V. Arhangel skiȋ, J. van Mill and G. J. Ridderbos in [2].…”

“…This was brought to my attention by F. D. Tall at the 2006 Prague Topological Symposium. Below we answer this question by extending some results from [6] to power homogeneous T 5 compacta. This partially solves [3,Problem 3.17].…”

“…As in [6], a compactum is an infinite compact space. By ∆(X, µ) we denote the diagonal in the space X µ , which consists of all points x ∈ X µ such that x α = x β for all α, β ∈ µ.…”