135 research outputs found
On paratopological groups
In this paper, we firstly construct a Hausdorff non-submetrizable
paratopological group in which every point is a -set, which
gives a negative answer to Arhangel'ski\v{\i}\ and Tkachenko's question
[Topological Groups and Related Structures, Atlantis Press and World Sci.,
2008]. We prove that each first-countable Abelian paratopological group is
submetrizable. Moreover, we discuss developable paratopological groups and
construct a non-metrizable, Moore paratopological group. Further, we prove that
a regular, countable, locally -paratopological group is a discrete
topological group or contains a closed copy of . Finally, we
discuss some properties on non-H-closed paratopological groups, and show that
Sorgenfrey line is not H-closed, which gives a negative answer to
Arhangel'ski\v{\i}\ and Tkachenko's question [Topological Groups and Related
Structures, Atlantis Press and World Sci., 2008]. Some questions are posed.Comment: 14 page
A note on rectifiable spaces
In this paper, we firstly discuss the question: Is
homeomorphic to a rectifiable space or a paratopological group? And then, we
mainly discuss locally compact rectifiable spaces, and show that a locally
compact and separable rectifiable space is -compact, which gives an
affirmative answer to A.V. Arhangel'ski\v{i} and M.M. Choban's question [On
remainders of rectifiable spaces, Topology Appl., 157(2010), 789-799]. Next, we
show that a rectifiable space is strongly Frchet-Urysohn if and
only if is an -sequential space. Moreover, we discuss the
metrizabilities of rectifiable spaces, which gives a partial answer for a
question posed in \cite{LFC2009}. Finally, we consider the remainders of
rectifiable spaces, which improve some results in \cite{A2005, A2007, A2009,
Liu2009}.Comment: 16 page
Dense-separable groups and its applications in -independence
A topological space is called {\it dense-separable} if each dense subset of
its is separable. Therefore, each dense-separable space is separable. We
establish some basic properties of dense-separable topological groups. We prove
that each separable space with a countable tightness is dense-separable, and
give a dense-separable topological group which is not hereditarily separable.
We also prove that, for a Hausdorff locally compact group , it is locally
dense-separable iff it is metrizable.
Moreover, we study dense-subgroup-separable topological groups. We prove
that, for each compact torsion (or divisible, or torsion-free, or totally
disconnected) abelian group, it is dense-subgroup-separable iff it is
dense-separable iff it is metrizable.
Finally, we discuss some applications in -independent topological groups
and related structures. We prove that each regular dense-subgroup-separable
abelian semitopological group with is
-independent. We also prove that, for each regular dense-subgroup-separable
bounded paratopological abelian group with , it is -independent
iff it is a nontrivial -group iff each nontrivial primary component
of is -independent. Apply this result, we prove that a separable
metrizable almost torsion-free paratopological abelian group with
is -independent. Further, we prove that each
dense-subgroup-separable MAP abelian group with a nontrivial connected
component is also -independent.Comment: 19 page
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