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