A Common Extension of Arhangel’skĭ’s Theorem and the Hajnal–Juhász Inequality

Abstract

We present a bound for the weak Lindelöf number of the G δ -modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: |X| ≤ 2 L(X)χ(X) (Arhangel'skiȋ) and |X| ≤ 2 c(X)χ(X) (Hajnal-Juhász). This solves a question that goes back to Bell, Ginsburg and Woods [6] and is mentioned in Hodel's survey on Arhangel'skiȋ's Theorem [15]. In contrast to previous attempts we do not need any separation axiom beyond T 2 .

Cardinal estimates involving the weak Lindelöf game

“…Bell, Ginsburg and Woods's main motivation to pose Question 1.1 was to find, at least within the realm of regular spaces, a common generalization to Arhangel'skii's Theorem and the Hajnal-Juhász's inequality stating that every first-countable space with the countable chain condition has cardinality at most continuum. Such a generalization was eventually found, even without assuming regularity, by the second and third author in [3].…”

A useful common weakening of Lindelöf and H-closed

“…A positive answer to the weaker question concerning only the couple Lindelöf and H-closed is given in [2] and [4]. By combining the results from [2] and [3], in this short note we present a property P which provides a full positive answer to Question 1.…”

Power homogeneous compacta and variations on tightness

“…This invariant has the properties pwL c (X) ≤ L(X) and pwL c (X) ≤ c(X). It was shown in [5] that if X is Hausdorff then |X| ≤ 2 pwLc(X)χ(X) , unifying the well-known bounds 2 L(X)χ(X) and 2 c(X)χ(X) for the cardinality of a Hausdorff space X. We show in Theorem 4.4 that if X is additionally homogeneous then |X| ≤ 2 pwLc(X)wt(X)πχ(X)pct(X) .…”

“…We use the cardinal invariant pwL c (X), first defined in [5] (see Definition 4.2 below). This invariant has the properties pwL c (X) ≤ L(X) and pwL c (X) ≤ c(X).…”