Weak developments and metrization

Abstract

The notions of a weak k-development and of a weak development, defined in terms of sequences of open covers, were recently introduced by the first and the third authors. The first notion was applied to extend in an interesting way Michael's Theorem on double set-valued selections. The second notion is situated between that of a development and of a base of countable order. To see that a space with a weak development has a base of countable order, we use the classical works of H.H. Wicke and J.M. Worrell. We also introduce and study the new notion of a sharp base, which is strictly weaker than that of a uniform base and strictly stronger than that of a base of countable order and of a weakly uniform base, and which is strongly connected to the notion of a weak development. Several examples are exhibited to prove that the new notions do not coincide with the old ones. In short, our results show that the notions of a weak development and of a sharp base fit very well into already existing system of generalized metrizability properties defined in terms of sequences of open covers or bases. Several open questions are formulated.