Glasnik Matematicki, Vol. 49, No. 2 (2014), 433-446.

DISCRETE REFLEXIVITY IN GO SPACES

Vladimir V. Tkachuk and Richard G. Wilson

Abstract.   A property P is discretely reflexive if a space X has P whenever Cl D has P for any discrete set D ⊂ X. We prove that quite a few topological properties are discretely reflexive in GO spaces. In particular, if X is a GO space and Cl D is first countable (paracompact, Lindelöf, sequential or Fréchet-Urysohn) for any discrete D ⊂ X then X is first countable (paracompact, Lindelöf, sequential or Fréchet-Urysoh 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 Čech-completeness is not discretely reflexive even in second countable linearly ordered spaces. However, discrete Čech-completeness of X × X is equivalent to its Čech-completeness if X is a LOTS. We also establish that any discretely Čech-complete Borel set must be Čech-complete.

2010 Mathematics Subject Classification.   54D45, 54F05, 54G12.

Key words and phrases.   Discretely reflexive property, discretely Lindelöf space, GO space, discretely locally compact space, discretely Čech-complete space, d-separable space, discretely scattered space, linearly ordered space.

DOI: 10.3336/gm.49.2.15

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