Salvador Garcia-Ferreira

Resolvability: A selective survey and some new results

Abstract Following guidance from the Organizing Committee, the authors give a brief introduction to the theory of spaces which are resolvable in the sense introduced by Hewitt (1943) . The new results presented here are these. (A) A countably compact regular Hausdorff space without isolated points is ω -resolvable—that is, it admits an infinite family of pairwise disjoint dense subsets. (B) Among Tychonoff topologies without isolated points on a fixed set, no pseudocompact topology is maximal.

Three orderings on β (ω)⧹ω

Abstract By using Bernsteins concept of p-compactness for pϵω∗, W.W. Comfort has defined the following pre-order on β(ω)⧹ω: for p,qϵβ(ω)⧹ω,p⩽cq if every q-compact space is p-compact. We showthat ⩽C≠⩽RK(⩽RK denotes the Rudin-Keisler order); ⩽C and ⩽RK coincide on the set of weakP-points of ω∗; for every pϵω∗ the set TC(p)={q:p⩽Cq and q⩽Cp} can be filled out with exactly 2ω types; if p is RK-minimal then (TC(p), ⩽RF) is a linearly ordered set (⩽RF= Rudin-Frolik order); ∃p,qϵω∗ such that p and q are RK-incomparable and p⩽Cq⩽Cp;⩽C may be defined in terms of ⩽RK; (∀p,q ϵω∗)(∃rϵω∗)[r⩽RKp∧r⩽RKq]⇔(∀p,qϵω∗) (∃rϵω∗)[r⩽Cp∧r⩽Cq]; for pϵω∗, TC(p) is countably compact; and ifp is a P-point then TC(p) is p-compact. Several open problems related to ⩽C are listed.

Dense subsets of maximally almost periodic groups

A (discrete) group G is said to be maximally almost periodic if the points of G are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology T on a group G is totally bounded if whenever ∅ 6= U ∈ T there is F ∈ [G] |G|, (b) each D ∈ D is dense in G, and (c) distinct D,E ∈ D satisfy |D ∩ E| < d(G); a totally bounded topological group with such a family is a strongly extraresolvable topological group. We give two theorems, the second generalizing the first. Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup. Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable. Theorem 2. Let G be maximally almost periodic. Then there are a subgroup H of G and a family D ⊆ P(H) such that (i) H is dense in every totally bounded group topology on G; (ii) the family D is a strongly extraresolvable family for every totally bounded group topology T on H such that d(H, T ) = |H|; and (iii) H admits a totally bounded group topology T as in (ii). Remark. In certain cases, for example when G is Abelian, one must in Theorem 2 choose H = G. In certain other cases, for example when the largest totally bounded group topology on G is compact, the choice H = G is impossible.

Some Generalizations of Pseudocompactness

ABSTRACT: In this paper, we introduce the concepts of p‐boundedness for pɛω*, (α, M)‐pseudocompactness and (α, M)‐compactness, for a cardinal number α and Ø≠M⊆β(ω)ω. We prove that Xα is pseudocompact (respectively, countably compact) iff X is (α, M)‐pseudocompact (respectively, (α, M)‐compact), for some Ø≠M⊆β(ω)ω; the Rudin‐Keisler order on β(ω)ω can be defined in terms of p‐boundedness and p‐pseudocompactness; and if pɛβ(ω)ω then p is RK‐minimal (selective) iff the space ω∪T(p) is p‐pseudocompact, where T(p) is the type of p in β(ω)ω.

Baire spaces and Vietoris hyperspaces

We prove that if the Vietoris hyperspace CL(X) of all nonempty closed subsets of a space X is Baire, then all finite powers of X must be Baire spaces. In particular, there exists a metrizable Baire space X whose Vietoris hyperspace CL(X) is not Baire. This settles an open problem of R. A. McCoy stated in 1975.

p-Fréchet-Urysohn property of function spaces

In this paper, we study the p-FrCchet-Urysohn property of function spaces, for p E /3(w)w. We prove that C,(X) is p-FrCchet-Urysohn if and only if X has (r,), where (7,) is the natural p-version of property (y) (this is a generalization of a result due to Gerlits and Nagy). We note the following implications: X is second countable *X has (7,) for some p EP(o)o -X n is LindelGf for all 1 Q n w such that C,(R) is p-FrCchet-Urysohn; if p is semiselective, then every subset X of R satisfying (r,,) has measure zero and if p is selective, then X is a strong measure zero set; and we can find p E /3(o>o such that C,(R) is p-FrCchet-Urysohn and is not strongly p-FrCchet-Urysohn. Finally, we prove that [w” does not have (-y,) whenever p is a P-point of P(w)w.

The α-Boundification of &#945

A space X is < α-BOUNDED IF FOR ALL A ⊆ X WITH |A| < α, CL X A is compact. Let B(α) be the smallest < α-bounded subspace of β(α) containing α. It is shown that the following properties are equivalent: (a) α is a singular cardinal; (b) B(α) is not locally compact; (c) B(α) is α-pseudocompact; (d) B(α) is initially α-compact. Define B 0 (α) = α and B n (α) = {cl β(α) A: A ⊆ B n−1 (α), |A| < α} FOR 0 < N < ω. WE ALSO PROVE THAT B 2 (α) ¬= B 3 (α) when ω = cf(α) < α. Finally, we calculate the cardinality of B(α) and prove that, for every singular cardinal α, |B(α)| = |B(α)| α = |N(α)| cf(α) where N(α) = {p ∈ β(α): there is A ∈ p with |A| < α}