Compatible algebraic structures on scattered compacta

Abstract

It is proved that each hereditarily collectionwise Hausdorff compact scattered space with finite scattered height admits a continuous semilattice operation turning it into a topological semilattice with open principal filters. On the other hand a compactification γ N of a countable discrete space N whose remainder is homeomorphic to [ 0 , ω 1 ] admits no (separately) continuous binary operation turning γ N into an inverse semigroup (semilattice). Also we construct a compactification ψ N of N admitting no separately continuous semilattice operation and such that the remainder ψ N ∖ N is homeomorphic to the one-point compactification of an uncountable discrete space. To show that ψ N admits no continuous semilattice operation we prove that the set of isolated points of a compact scattered topological semilattice X of scattered height 2 is sequentially dense in X. Also we prove that each separable scattered compactum with scattered height 2 is a subspace of a separable compact scattered topological semilattice with open principal filters and scattered height 2. This allows us to construct an example of a separable compact scattered topological semilattice with open principal filters and scattered height 2, which fails to be Fréchet–Urysohn. Also we construct an example of a Fréchet–Urysohn separable non-metrizable compact scattered topological semilattice with open principal filters and scattered height 2.