Pinning down versus density

Juhász, István and Soukup, Lajos and Szentmiklóssy, Zoltán (2016) Pinning down versus density. ISRAEL JOURNAL OF MATHEMATICS, 215 (2). pp. 583-605. ISSN 0021-2172

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Abstract

The pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for any neighborhood assignment U: X → τX there is a set A ∈ [X]κwith A ∩ U(x) ≠ Ø for all x ∈ X. Clearly, c(X) ≤ pd(X) ≤ d(X). Here we prove that the following statements are equivalent (1) 2κ< κ+ωfor each cardinal κ (2) d(X) = pd(X) for each Hausdorff space X (3) d(X) = pd(X) for each 0-dimensional Hausdorff space X. This answers two questions of Banakh and Ravsky. The dispersion character Δ(X) of a space X is the smallest cardinality of a non-empty open subset of X. We also show that if pd(X) < d(X) then X has an open subspace Y with pd(Y) < d(Y) and |Y| = Δ(Y), moreover the following three statements are equiconsistent (i) There is a singular cardinal λ with pp(λ) > λ+, i.e., Shelah’s Strong Hypothesis fails (ii) there is a 0-dimensional Hausdorff space X such that |X| = Δ(X) is a regular cardinal and pd(X) < d(X) (iii) there is a topological space X such that |X| = Δ(X) is a regular cardinal and pd(X) < d(X). We also prove that • d(X) = pd(X) for any locally compact Hausdorff space X • for every Hausdorff space X we have |X| ≤ 22pd(x) and pd(X)<d(X) implies Δ(X) <22pd(x) • for every regular space X we have min{Δ(X), w(X)} ≤ 2pd(X)and d(X) < 2pd(X), moreover pd(X) < d(X) implies Δ(X) < 2pd(X). © 2016, Hebrew University of Jerusalem.

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