Topology Proceedings 3 (1978) pp. 237-265: DISCRETE SEQUENCES OF POINTS

We consider a weak version of R. L. Moore's property D. Roughly speaking, a space X is said to have property D if each discrete (in the locally finite sense) sequence of points in X can be "expanded" to a discrete family of open sets in X. A space is said to have property wD if each discrete sequence has a subsequence which can be "expanded" to a discrete family of open sets. All regular, submetrizable spaces and all realcompact spaces have property wD. In the class of regular spaces in which every point is a Go' propert wD is both hereditary and wI-fold productive. In the class of T3-spaces, finite to-one perfect maps preserve property D, but do not necessarily preserve property wD (for example, we show that certain finite-to-one perfect images of the Niemytzski plane and of the Pixley-Roy space do not have property wD). Property wD, however, is preserved by n-to-one perfect maps for every positive integer n. Whether every product of perfectly normal T1-spaces has property wD is independent of the usual axioms of set theory.