Resolvability of spaces having small spread or extent

In a recent paper O. Pavlov proved the following two interesting resolvability results:(1)If a T1-space X satisfies Δ(X)>ps(X) then X is maximally resolvable.(2)If a T3-space X satisfies Δ(X)>pe(X) then X is ω-resolvable.Here ps(X) (pe(X)) denotes the smallest successor cardinal such that X has no discrete (closed discrete) subset of that size and Δ(X) is the smallest cardinality of a non-empty open set in X.In this note we improve (1) by showing that Δ(X)>ps(X) can be relaxed to Δ(X)⩾ps(X), actually for an arbitrary topological space X. In particular, if X is any space of countable spread with Δ(X)>ω then X is maximally resolvable.The question if an analogous improvement of (2) is valid remains open, but we present a proof of (2) that is simpler than Pavlov's.