S. Garcia-Ferreira

Countably compact groups from a selective ultrafilter

We prove that the existence of a selective ultrafilter on ω implies the existence of a countably compact group without non-trivial convergent sequences all of whose powers are countably compact. Hence, by using a selective ultrafilter on ω, it is possible to construct two countably compact groups without non-trivial convergent sequences whose product is not countably cornpact.

Extreme selections for hyperspaces of topological spaces

Abstract We study properties of Hausdorff spaces X which depend on the variety of continuous selections for their Vietoris hyperspaces F (X) of closed non-empty subsets. Involving extreme selections for F (X) , we characterize several classes of connected-like spaces. In the same way, we also characterize several classes of disconnected-like spaces, for instance all countable scattered metrizable spaces. Further, involving another type of selections for F (X) , we study local properties of X related to orderability. In particular, we characterize some classes of orderable spaces with only one non-isolated point.

On Cα-compact subsets

Abstract For an infinite cardinal α, we say that a subset B of a space X is Cα-compact in X if for every continuous function f : X → Rα, [B] is a compact subset of Rα. This concept slightly generalizes the notion of α-pseudocompactness introduced by J.F. Kennison: a space X is α-pseudocompact ifX is Cα-compact in itself. If α = ω, then we say C-compact instead of Cω-compact and ω-pseudocompactness agrees with pseudocompactness. We generalize Tamanos theorem on the pseudocompactness of a product of two spaces as follows: let A ⊆ X and B ⊆ Y be such that A is z-embedded in X. Then the following three conditions are equivalent: (1) A × B is Cα-compact in X × Y; (2) A and B are Cα-compact in X and Y, respectively, and the projection map π : X × Y → X is a zα-map with respect to A × B and A; and (3) A and B are Cω-compact in X and Y, respectively, and the projection map π : X × Y → X is a strongly zα-map with respect to A × B and A (the zα-maps are the strongly zα-maps are natural generalizations of the z-maps and the strongly z-maps, respectively). The degree of Cα-compactness of a C-compact subset B of a space X is defined by: ϱ(B,X) = ∞ if B is compact, and if B is not compact, then ϱ(B,X) = supα: B is Cα-compact in X. We estimate the degree of pseudocompactness of locally compact pseudocompact spaces, topological products and Σ-products. We also establish the relation between the pseudocompact degree and some other cardinal functions. In the context of uniform spaces, we show that if A is a bounded subset of a uniform space (X,U), then A is Cα-compact in X , where ( X , U ), is the completion of (X,U) iff f(A) is a compact subset of Rα from every uniformly continuous function from X into Rα; we characterize the Cα-compact subsets of topological groups; and we also prove that if Gi: i I is a set of topological groups 15 and Ai is a Cα-compact subset of Gα for all i I, then ΠiI Ai is a Cα-compact subset of ΠiI Gi.

Some remarks on extraresolvable spaces

Abstract We give an example of a countable extraresolvable space that is not strongly extraresolvable. We also prove that Q ×ω 1 is strongly extraresolvable, and if X is strongly extraresolvable and nwd(X)=Δ(X)=ω, then X×ω is strongly extraresolvable (ω1 and ω are equipped with the discrete topology).

Topological games defined by ultrafilters

Abstract By using a free ultrafilter p on ω , we introduce an infinite game, called G p (x,X) -game, played around a point x in a space X . This game is the natural generalization of the G (x,X) -game introduced by A. Bouziad. We establish some relationships between the G p (x,X) -game and the Rudin–Keisler pre-order on ω ∗ . We prove that if p,q∈ω ∗ , then βω ⧹ P RK ( p ) is a G q -space if and only if q≰ RK p ; and, for every p∈ω ∗ , there is a G p -space that is not a G q -space for every q ∈ T ( p )⧹ R ( p ), where R(p)={ f (p): ∃A∈p(f| A is strictly increasing )} . As a consequence, we characterize the Q -points in ω ∗ as follows: p∈ω ∗ is a Q -point iff every G p -space is a G q -space for every q ∈ T ( p ), where T(p)={q∈ω ∗ : p⩽ RK q and q⩽ RK p} .