A note on discrete sets

We give several partial positive answers to a question of Juhasz and Szentmiklossy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences and their closures with the cardinality of a Hausdorff space, improving known results in the literature.

💡 Research Summary

The paper investigates the relationship between the minimal number of discrete subsets needed to cover a compact space, denoted dis(X), and the smallest cardinality of a non‑empty open set, denoted Δ(X). This question originates from Juhász and Szentmiklossy’s problem: does every compact space X satisfy dis(X) ≥ Δ(X)? The author provides several partial positive answers for a wide variety of compact and related spaces, and also explores extensions beyond compactness.

The first part of the paper builds on Gruenhage’s theorem that a perfect map does not increase dis(·). Using a lemma about cellular families, the author shows that if every open set contains a cellular family of size κ, then dis(X) ≥ κ^ω. Consequently, for any hereditarily collectionwise Hausdorff compact space, dis(X) ≥ Δ(X) (Theorem 2.5). Since compact linearly ordered topological spaces (LOTS) are monotonically normal and thus hereditarily collectionwise Hausdorff, the same inequality holds for compact LOTS (Corollary 2.6). Under the constructibility axiom V = L, all compact T₅ spaces are hereditarily collectionwise Hausdorff, giving the result in that model; a weaker set‑theoretic hypothesis (2^κ < 2^{κ+}) also suffices (Theorem 2.7).

The paper then turns to special classes of compact spaces. For polyadic compacta (continuous images of powers of one‑point compactifications of discrete sets) Lemma 2.11 and Lemma 2.12 give the equality w(P) = t(P)·c(P). Using these, Theorem 2.13 proves dis(X) ≥ Δ(X) for any polyadic compactum. Gul’ko compacta, which are known to contain a dense Baire metrizable subspace, are treated via meta‑Lindelöfness. Lemma 2.14 shows that in a hereditarily meta‑Lindelöf space any subset of size ≤ κ has cardinality ≤ κ, and Theorem 2.15 deduces dis(X) ≥ Δ(X) for any hereditarily meta‑Lindelöf space with a dense Baire metrizable subset. Consequently every Gul’ko compactum satisfies the inequality (Corollary 2.16).

The author also studies Σ‑products of Cantor cubes. By constructing an adequate family of chains in a specially ordered set, an Eberlein compactum X with Δ(X) = κ^ω is embedded into Σ(2^κ). Theorem 2.19 then yields dis(Σ(2^κ)) = κ^ω, showing that even non‑compact countably compact Σ‑products obey a similar lower bound.

The final section focuses on homogeneous and power‑homogeneous compacta. Lemma 2.24 (Juhász–van Mill) guarantees a point of character < dis(X) in any infinite compact space. Lemmas 2.25 and 2.26 relate dense sets of points of small π‑character to the overall π‑character and character of the space. Lemma 2.27 proves that for power‑homogeneous compacta, Δ(X) ≤ 2·dis(X). Under GCH, Theorem 2.28 shows dis(X) ≥ min{Δ(X), ℵ_ω}; under CH, Theorem 2.30 improves this to dis(X) ≥ min{Δ(X), ω₃}. Corollary 2.29 deduces that any power‑homogeneous compactum of size ≤ ℵ_ω satisfies dis(X) ≥ Δ(X).

Throughout, the paper refines known cardinal inequalities (e.g., dis(X) ≥ c, dis(X) ≥ 2^κ) and demonstrates that the lower bound Δ(X) is often attainable. The author also lists open problems, notably whether the inequality holds for all Corson compacta or for any compact space possessing a dense Baire metrizable subspace.

In summary, the work substantially advances our understanding of how discrete families reflect the cardinal structure of compact (and some non‑compact) spaces, providing a unified framework that covers many classical classes and suggesting further directions for set‑theoretic topology.