Abstract
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. Yajima's 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.
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