Glasnik Matematicki, Vol. 49, No. 2 (2014), 433446.
DISCRETE REFLEXIVITY IN GO SPACES
Vladimir V. Tkachuk and Richard G. Wilson
Departamento de Matemáticas,
Universidad Autónoma Metropolitana,
Av. San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa,
C.P. 09340, Mexico D.F.,
Mexico
email: vova@xanum.uam.mx
email: rgw@xanum.uam.mx
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 FréchetUrysohn) for any discrete D ⊂ X then X
is first countable (paracompact, Lindelöf, sequential or FréchetUrysoh
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
Čechcompleteness is not discretely reflexive even in second
countable linearly ordered spaces. However, discrete Čechcompleteness of X × X is equivalent to its Čechcompleteness if X is a LOTS. We also establish that any
discretely Čechcomplete Borel set must be Čechcomplete.
2010 Mathematics Subject Classification.
54D45, 54F05, 54G12.
Key words and phrases. Discretely reflexive property, discretely
Lindelöf space, GO space, discretely locally compact space,
discretely Čechcomplete space, dseparable space,
discretely scattered space, linearly ordered space.
Full text (PDF) (access from subscribing institutions only)
DOI: 10.3336/gm.49.2.15
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