Original scientific paper

https://doi.org/10.3336/gm.49.2.15

Discrete reflexivity in GO spaces

Vladimir V. Tkachuk ; Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa, C.P. 09340, Mexico D.F., Mexico
Richard G. Wilson ; Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa, C.P. 09340, Mexico D.F., Mexico

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.

Keywords

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

Hrčak ID:

130895

URI

https://hrcak.srce.hr/130895

Publication date:

18.12.2014.

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