On weakening tightness to weak tightness

Abstract

The weak tightness wt(X) of a space X was introduced in [11] with the property wt(X) ≤ t(X). We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that wt(X) = ℵ0 < t(X) under 2 ℵ 0 = 2 ℵ 1 . In particular, this demonstrates the celebrated Balogh's Theorem [5] does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then |X| ≤ 2 wt(X)πχ (X) . This refines a theorem of De la Vega [12]. In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhász and van Mill [15]. Third, we show that if X is a T1 space, wt(X) ≤ κ, X is κ +compact, and ψ(D, X) ≤ 2 κ for any D ⊆ X satisfying |D| ≤ 2 κ , then a) d(X) ≤ 2 κ and b) X has at most 2 κ -many Gκ-points. This is a variation of another theorem of Balogh [6]. Finally, we show that if X is a regular space, κ = L(X)wt(X), and λ is a caliber of X satisfying κ < λ ≤ (2 κ ) + , then d(X) ≤ 2 κ . This extends of theorem of Arhangel ′ skiȋ [3].

Power homogeneous compacta and variations on tightness

“…We introduce the almost tightness at(X) with the property wt(X) ≤ at(X) ≤ t(X) and show that if X is a power homogeneous compactum then |X| ≤ 2 at(X)πχ(X) . This improves the result of Arhangel ′ skiȋ, van Mill, and Ridderbos in [2] that |X| ≤ 2 t(X) for a power homogeneous compactum X and gives a partial answer to a question in [4]. In addition, if X is a homogeneous Hausdorff space we show that |X| ≤ 2 pwc L(X)wt(X)πχ(X)pct(X) , improving a result in [3].…”

“…(See 3.14 for the definition of pct(X) and 3.3 for the definition of C-saturated). This represents an extension of Theorem 3.3 in [4] into the Hausdorff setting. That theorem established the existence of G and H with the above properties when X is compact and wt(X) ≤ κ.…”

“…Motivated by results of Juhász and van Mill in [13], the weak tightness wt(X) of a topological space X was introduced in [6] with the property wt(X) ≤ t(X) (see Definition 3.1). This cardinal invariant was investigated further by Bella and the author in [4], where the following theorem was shown. Recall a space X is homogeneous if for all x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. X is power homogeneous if there exists a cardinal κ such that X κ is homogeneous.…”

“…It was asked in Question 3.13 in [4] if 2 wt(X)πχ(X) is a bound for the cardinality of a power homogeneous compactum X. While this question is still open, we attain a partial answer by introducing a cardinal invariant related to wt(X), the almost tightness at(X) of a space X (see Definition 3.2).…”

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The weak tightness wt(X), introduced in [6], has the property wt(X) ≤ t(X). It was shown in [4] that if X is a homogeneous compactum then |X| ≤ 2 wt(X)πχ(X) . We introduce the almost tightness at(X) with the property wt(X) ≤ at(X) ≤ t(X) and show that if X is a power homogeneous compactum then |X| ≤ 2 at(X)πχ(X) .

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