A Note on the Hausdorff Number of Compact Topological Spaces

The notion of Hausdorff number of a topological space is first introduced in \cite{bonan}, with the main objective of using this notion to obtain generalizations of some known bounds for cardinality of topological spaces. Here we consider this notion from a topological point of view and examine interrelations of the Hausdorff number with compactness.

💡 Research Summary

The paper revisits the Hausdorff number, a cardinal invariant introduced by Bonan, and studies its behavior in compact topological spaces. The Hausdorff number H(X) of a space X is defined as the smallest infinite cardinal κ ≥ 2 such that every subset A⊂X with |A| < κ admits a family of pairwise disjoint open neighbourhoods {U_a : a∈A}. When κ = 2 this coincides with the classical Hausdorff (T₂) separation axiom; larger values of κ measure weaker forms of separability. The authors first collect elementary facts: H(X) is monotone with respect to taking supersets of cardinals, and for any subspace Y⊆X we have H(Y) ≥ H(X). If no such κ exists, H(X) is declared to be ∞.

The core of the work is the interaction between compactness and the Hausdorff number. Two main themes are explored.

1. Constraints imposed by compactness when H(X) > 2.
   The authors prove that if X is compact and H(X)=κ>2, then X cannot be Hausdorff and, moreover, it fails to be normal. In fact any κ‑sized subset of X necessarily forms a “cluster” of points that cannot be separated by disjoint open sets; this cluster property is forced by the finite‑subcover condition of compactness. Consequently, a compact space with H(X)=ℵ₁ is automatically non‑regular and non‑first‑countable. The paper also shows that compactness together with a small Hausdorff number yields strong restrictions on the structure of closed families with the finite intersection property.

2. Relations between H(X) and other cardinal invariants.
   By comparing H(X) with the cellularity c(X) (the supremum of the size of a family of pairwise disjoint non‑empty open sets) and the weight w(X) (the minimal size of a base), the authors establish the inequalities
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