DISCRETE REFLEXIVITY IN GO SPACES

A property P is discretely reflexive if a space X has P whenever (D) over bar has P for any discrete set D subset of X. We prove that quite a few topological properties are discretely reflexive in GO spaces. In particular, if X is a GO space and (D) over bar is first countable (paracompact, Lindelof, sequential or Frechet-Urysohn) for any discrete D subset of X then X is first countable (paracompact, Lindelof, sequential or Frechet-Urysohn respectively). We show that a space with a nested local base at every point is discretely locally compact if and only if it is locally compact. Therefore local compactness is discretely reflexive in GO spaces. It is shown that a GO space is scattered if and only if it is discretely scattered. Under CH we show that Cech-completeness is not discretely reflexive even in second countable linearly ordered spaces. However, discrete Cech-completeness of X x X is equivalent to its Cech-completeness if X is a LOTS. We also establish that any discretely Cech-complete Borel set must be Cech-complete.