On $\sigma $-countably tight spaces

Abstract

Abstract. Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality c if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and σ-countably tight compactum has cardinality c remains open.We also show that if an arbitrary product is σ-countably tight then all but finitely many of its factors must be countably tight.

On weakening tightness to weak tightness

“…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.…”

“…In Theorem 4.1 a closing-off argument is used to show that if X is T 1 , κ is a cardinal, wt(X) ≤ κ, X is κ + -compact, and ψ(D, X) ≤ 2 κ for any D ⊆ X satisfying |D| ≤ 2 κ , then d(X) ≤ 2 κ and there are at most 2 κ -many G κ -points. This result is related to Lemma 3.2 in [15] and produces an alternative proof that |X| ≤ 2 L(X)wt(X)ψ(X) for a Hausdorff space X.…”

“…The goal in this section is to give an improved bound for the cardinality of any homogeneous compactum. For a space X, the notion of a C-saturated subset of a space X was introduced in [15]. Given a cover C of X,…”

“…In Theorem 2.5 in [15], Juhász and van Mill showed that if a compact space X is a countable union of countable tight subspaces then there is a non-empty G δ -set contained in the closure of a countable set. That is, a σ-CT compact space has a non-empty subseparable G δ -set.…”

“…The cardinal function t(X), the tightness of a topological space X, is the least infinite cardinal κ such that whenever A ⊆ X and x ∈ A then there exists B ⊆ A such that |B| ≤ κ and x ∈ B. Motivated by results of Juhász and van Mill in [15], the cardinal function wt(X), the weak tightness of X, was introduced in [11]. (See Definition 2.1 below).…”