78 research outputs found

### A note on free paratopological groups

In this paper, we mainly discuss some generalized metric properties and the
character of the free paratopological groups, and extend several results valid
for free topological groups to free paratopological groups.Comment: 12 page

### Topologically subordered rectifiable spaces and compactifications

A topological space $G$ is said to be a {\it rectifiable space} provided that
there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and
an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in
G$ we have $\phi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ is the
projection to the first coordinate. In this paper, we mainly discuss the
rectifiable spaces which are suborderable, and show that if a rectifiable space
is suborderable then it is metrizable or a totally disconnected P-space, which
improves a theorem of A.V. Arhangel'ski\v\i\ in \cite{A20092}. As an
applications, we discuss the remainders of the Hausdorff compactifications of
GO-spaces which are rectifiable, and we mainly concerned with the following
statement, and under what condition $\Phi$ it is true.
Statement: Suppose that $G$ is a non-locally compact GO-space which is
rectifiable, and that $Y=bG\setminus G$ has (locally) a property-$\Phi$. Then
$G$ and $bG$ are separable and metrizable.
Moreover, we also consieder some related matters about the remainders of the
Hausdorff compactifications of rectifiable spaces.Comment: 14 pages (replace

### Local properties on the remainders of the topological groups

When does a topological group $G$ have a Hausdorff compactification $bG$ with
a remainder belonging to a given class of spaces? In this paper, we mainly
improve some results of A.V. Arhangel'ski\v{\i} and C. Liu's. Let $G$ be a
non-locally compact topological group and $bG$ be a compactification of $G$.
The following facts are established: (1) If $bG\setminus G$ has a locally a
point-countable $p$-metabase and $\pi$-character of $bG\setminus G$ is
countable, then $G$ and $bG$ are separable and metrizable; (2) If $bG\setminus
G$ has locally a $\delta\theta$-base, then $G$ and $bG$ are separable and
metrizable; (3) If $bG\setminus G$ has locally a quasi-$G_{\delta}$-diagonal,
then $G$ and $bG$ are separable and metrizable. Finally, we give a partial
answer for a question, which was posed by C. Liu in \cite{LC}.Comment: 10pages (replace

### On paratopological groups

In this paper, we firstly construct a Hausdorff non-submetrizable
paratopological group $G$ in which every point is a $G_{\delta}$-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 $k_{\omega}$-paratopological group is a discrete
topological group or contains a closed copy of $S_{\omega}$. 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

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