$G_δ$ covers of compact spaces

Abstract

We solve a long standing question due to Arhangel'skii by constructing a compact space which has a G δ cover with no continuum-sized (G δ )-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every G δ cover has a c-sized subcollection with a G δdense union and that in a Lindelöf space with a base of multiplicity continuum, every G δ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.

Cardinal Invariants for the $G_\delta $ topology

“…At the 1970 International Congress of Mathematicians in Nice, France, Arhangel'skii asked whether the weak Lindelöf degree of a compact space with its G δ topology is always bounded by the continuum. A counterexample has recently been given in [22] but various related bounds for the (weak) Lindelöf number of the G δ topology have been presented in the literature (see, for example [12], [20], [14] and [7]).…”

“…There are various papers in the literature investigating what properties of X are preserved when passing to X δ and presenting bounds for cardinal invariants on X δ in terms of the cardinal invariants of X (see for example [14], [12], [20], [17]). Moreover, results of that kind have found applications to central topics in general topology like the study of covering properties in box products (see, for example, [18]), cardinal invariants for homogeneous compacta (see, for example [2], [6], [7] and [22]) and spaces of continuous functions (See [1]).…”

“…Two of the early results on this topic are Juhász's bound c(X δ ) ≤ 2 c(X) for every compact Hausdorff space X, where c(X) denotes the cellularity of X and Arhangel'skii's result that the G δ topology on a Lindelöf regular scattered space is Lindelöf. Juhász's bound is tight in the sense that it's not possible to prove that c(X δ ) ≤ c(X) ω for every compact space X (see [11]) and the scattered property is essential in Arhangel'skii's result because there are compact Hausdorff spaces whose G δ -topology even has (weak) Lindelöf number c + (see [22]).…”

A definitive improvement of a game-theoretic bound and the long tightness game

“…This seems to suggest a possible further strengthening of Theorem 5 as follows: if X is a regular space where Bob has a winning strategy in G ω1 fin (O, O), then L(X δ ) ≤ 2 ω . However, this conjecture drastically fails because there are compact T 2 spaces such that the Lindelöf degree of the G δ -modification is much bigger than the continuum (see e. g. [11] or [12]), while for every compact space Bob may win in G ω1 fin (O, O) at the first inning!…”