When is an ultracomplete space almost locally compact?

[EN] We study spaces X which have a countable outer base in βX; they are called ultracomplete in the most recent terminology. Ultracompleteness implies Cech-completeness and is implied by almost local compactness (≡having all points of non-local compactness inside a compact subset of countable outer character). It turns out that ultracompleteness coincides with almost local compactness in most important classes of isocompact spaces (i.e., in spaces in which every countably compact subspace is compact). We prove that if an isocompact space X is ω-monolithic then any ultracomplete subspace of X is almost locally compact. In particular, any ultracomplete subspace of a compact ω-monolithic space of countable tightness is almost locally compact. Another consequence of this result is that, for any space X such that vX is a Lindelöf Σ-space, a subspace of Cp(X) is ultracomplete if and only if it is almost locally compact. We show that it is consistent with ZFC that not all ultracomplete subspaces of hereditarily separable compact spaces are almost locally compact.Research supported by Consejo Nacional de Ciencia y Tecnología (CONACyT) of Mexico grants 94897 and 400200-5-38164-E.Jardón Arcos, D.; Tkachuk, VV. (2006). When is an ultracomplete space almost locally compact?. Applied General Topology. 7(2):191-201. doi:10.4995/agt.2006.1923.SWORD1912017

When is an ultracomplete space almost locally compact?

We study spaces X which have a countable outer base in βX; they are called ultracomplete in the most recent terminology. Ultracompleteness implies Cech-completeness and is implied by almost local compactness (≡having all points of non-local compactness inside a compact subset of countable outer character). It turns out that ultracompleteness coincides with almost local compactness in most important classes of isocompact spaces (i.e., in spaces in which every countably compact subspace is compact). We prove that if an isocompact space X is ω-monolithic then any ultracomplete subspace of X is almost locally compact. In particular, any ultracomplete subspace of a compact ω-monolithic space of countable tightness is almost locally compact. Another consequence of this result is that, for any space X such that vX is a Lindelöf Σ-space, a subspace of Cp(X) is ultracomplete if and only if it is almost locally compact. We show that it is consistent with ZFC that not all ultracomplete subspaces of hereditarily separable compact spaces are almost locally compact

ONE MORE VARIATION OF THE POINT-OPEN GAME

Abstract. A topological game “Dense Gδσ-sets ” (also denoted by DG) is introduced as follows: for any n ∈ ω at the n-th move the player I takes a point xn ∈ X and II responds by taking a Gδ-set Qn in the space X such that xn ∈ Qn. The play stops after ω moves and I wins if the set ⋃ {Qn: n ∈ ω} is dense in X. Otherwise the player II is declared to be the winner. We study classes of spaces on which the player I has a winning strategy. It is evident that the I-favorable spaces constitute a generalization of the class of separable spaces. We show that there exists a neutral space for the game DG and prove, among other things, that Lindelöf scattered spaces and dyadic spaces are I-favorable. We characterize I-favorability for the game DG in the spaces Cp(X); one of the applications is that, for a Lindelöf Σ-space X, the space Cp(X) is I-favorable for DG if and only if X is ω-monolithic. 1