$${G_\delta}$$ G δ covers of compact spaces

On the weak tightness and power homogeneous compacta

“…Both follow from Pytkeev's theorem. Results in a similar vein occur in [5], [6], [8], [19], and elsewhere.…”

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]).…”