About DK-like spaces and some applications

Let CSK be the class of all K-scattered spaces having countable ranks. It is shown in this paper that if X is a regular θ-refinable space, then player one has a winning strategy in G(DK,X) if and only if he has one in G(CSK,X). This partly answers Y. Yajimaʹs problem: By topological games, I prove that hereditary disconnectedness, zero-dimensionality and strong zero-dimensionality are equivalent in the realm of non-empty normal compact-scattered weak θ-refinable spaces. A collectionwise normal ultraparacompact-like space is an ultraparacompact space.