Cardinal Invariants for the $G_\delta $ topology

Abstract

We prove upper bounds for the spread, the Lindelöf number and the weak Lindelöf number of the G δ topology on a topological space and apply a few of our bounds to give a short proof to a recent result of Juhász and van Mill regarding the cardinality of a σ-countably tight homogeneous compactum.2000 Mathematics Subject Classification. Primary: 54A25, Secondary: 54D20, 54G20.

On the weak tightness and power homogeneous compacta

“…Third, a result of Bella and Spadaro from [6] follows as a corollary to the Main Theorem and can be strengthened to the power homogeneous setting. In that paper Theorem 1.1 was extended using the G δ -modification X δ by showing that if X is a homogeneous compactum which is the union of countably my dense countably tight subspaces, then L(X δ ) ≤ c. Using the Main Theorem, we show in Theorem 4.7 that this result holds even if the homogeneity property is replaced with the weaker power homogeneity property.…”

On generalization of theorems of Pestryakov

“…Given a topological space X we can consider a finer topology on X by declaring countable intersections of open subsets (or zero-sets) of X to be a base (see [4,9]). The new space is called the G δ topology of X and is denoted with X ℵ 0 (X δ in paper [6]). Note that the topology X ℵ 0 coincides with the weak topology generated by B α (X) for each α > 0 ( [17]).…”

Free sequences and the tightness of pseudoradial spaces

“…Sometimes a neat ZFC bound is avaliable. For example, Juhász [9] proved that c(X δ ) ≤ 2 c(X) for every compact Hausdorff space X, where c(X) denotes the cellularity of X, Bella and the author [4] proved that s(X δ ) ≤ 2 s(X) for every Hausdorff space X, where s(X) is the spread of X and Carlson, Porter and Ridderbos [6] showed that L(X δ ) ≤ 2 F (X)•L(X) for every Hausdorff space X (the F (X) in the exponent cannot be removed in the last result as there are even compact spaces whose weak Lindelöf number is larger than the continuum, see [15]). In other cases the existence of a bound may strongly depend on your set theory.…”