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Transitivity and Ortho-Bases

Throughout this paper “space” means “T 1 topological space.“ 1. The concept of an ortho-base was introduced by W. F. Lindgren and P. J. Nyikos. Definition 1. A base of a space X is called an ortho-base provided that for each subcollection either is open or is a local base of a point x ∊ X [17]. Ortho-bases are related to interior-preserving collections which have been known for some time. Definition 2. A collection of open sets of a space X is called interior-preserving provided that the intersection of any subcollection is open. A space X is called orthocompact provided that each open cover has an open interior-preserving refinement. It was proved in [17], in particular, that each space with an ortho-base is orthocompact, and each orthocompact developable space (which is the same as a non-archimedean quasi-metrizable developable space [4]) has an ortho-base.