Pseudocompact Δ -spaces are often scattered

Given a pseudocompact Δ-space X, we establish that countable subsets of X must be scattered. This implies that pseudocompact Δ-spaces of countable tightness are scattered. If a pseudocompact Δ-space has the Souslin property, then it is separable and has a dense set of isolated points. It is shown that adding a countable subspace to a pseudocompact Δ-space can destroy the Δ-property. However, if X is countably compact and Y⊂ X is a Δ-space for some Y⊂ X such that | X\ Y| ≤ ω, then X is a Δ-space. We also show that monotonically normal Δ-spaces must be hereditarily paracompact. Besides, if X is a subspace of an ordinal with its order topology, then X is hereditarily paracompact if and only if it has the Δ-property.