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Every semi-stratifiable space or strong $\Sigma$-space has a $\sigma$-cushioned (mod$k$)-network. In this paper it is showed that every space with a $\sigma$-cushioned (mod$k$)-network is a D-space, which is a common generalization of some results about D-spaces.

This paper introduces partial tvs-cone metric spaces as a common generalization of both tvs-cone metric spaces and partial metric spaces, and gives a new fixed point theorem for contractions of Nadler type on partial tvs-cone metric spaces. As corollaries, we obtain the main results of S. B. Nadler (1969), D. Wardowski (2011), S. Radenović et al. (2011) and H. Aydi et al. (2012) are deduced.

In this paper, we prove that a space $X$ is a $g$-metrizable space if and only if $X$ is a weak-open, $\pi$ and $\sigma$-image of a semi-metric space, if and only if $X$ is a strong sequence-covering, quotient, $\pi$ and $mssc$-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”.

The concepts of $k$-systems, $k$-networks and $k$-covers were defined by A. Arhangel’skiǐ in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among $k$-systems, $k$-networks and $k$-covers are further discussed and are established by $mk$-systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of $mk$-systems.

In this paper, we give characterizations of certain weak-open images of metric spaces.

In this paper $ss$-quotient maps and $ssq$-spaces are introduced. It is shown that (1) countable tightness is characterized by $ss$-quotient maps and quotient maps; (2) a space has countable tightness if and only if it is a countably bi-quotient image of a locally countable space, which gives an answer for a question posed by F. Siwiec in 1975; (3) $ssq$-spaces are characterized as the $ss$-quotient images of metric spaces; (4) assuming $2^{\omega }<2^{{\omega }_1}$, a compact $T_2$-space is an $ssq$-space if and only if every countably compact subset...

Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group....

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $$\Delta u+\rho \left(\frac{e^u}{{\int }_T{e}^u}-\frac{1}{\left|T\right|}\right)=0,{\int }_{T}u=0.$$ It is well-known that the associated energy functional admits a minimizer for each $\rho \le 8\pi$. The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting ${\lambda }_1\left(T\right)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le min\{8\pi ,{\lambda }_1\left(T\right)|T\left|\right\}$. Our results hold more generally for solutions that are Steiner...

We consider a sequence of multi-bubble solutions $u_k$ of the following fourth order equation $${\Delta }^2{u}_k={\rho }_k\frac{h\left(x\right)e^{u_k}}{{\int }_{\Omega }h{e}^{u_k}}\text{in}\Omega ,u_k=\Delta u_k=0\text{on}\partial \Omega ,(*)$$ where $h$ is a $C^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^4$, and ${\rho }_k$ is a constant such that ${\rho }_k\le C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_k=32{\sigma }_{3}m$ for some positive integer $m\ge 1$, where ${\sigma }_3$ is the area of the unit sphere in ${ℝ}^4$. Furthermore, we obtain the following sharp estimates for ${\rho }_k$: $$\begin{array}{cc}\hfill {\rho }_k-32{\sigma }_{3}m& =c_0\sum _{j=1}^m{ϵ}_{k,j}^2\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\hfill \\ & +o\left(\sum _{j=1}^m{ϵ}_{k,j}^2\right)\hfill \end{array}$$ where $c_0\&gt;0$, $log\frac{64}{{ϵ}_{k,j}^4}=\underset{x\in B_{\delta }\left(p_j\right)}{max}{u}_k\left(x\right)-log\left(\underset{\Omega }{\int }h{e}^{u_k}\right)$ and $u_k\to 32{\sigma }_3\sum _{j=1}^m{G}_4(·,p_j)$ in $C_{\mathrm{loc}}^4(\Omega \setminus \{p_1,...,p_m\})$. This yields a bound of solutions as ${\rho }_k$ converges...

We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G/H$ is sequentially connected, then so is $G$.

The main purpose of this paper is to establish general conditions under which $T_2$-spaces are compact-covering images of metric spaces by using the concept of $cfp$-covers. We generalize a series of results on compact-covering open images and sequence-covering quotient images of metric spaces, and correct some mapping characterizations of $g$-metrizable spaces by compact-covering $\sigma$-maps and $mssc$-maps.

By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.

In this paper the relations of mappings and families of subsets are investigated in Ponomarev-systems, and the following results are obtained. (1) $f$ is a sequence-covering (resp. 1-sequence-covering) mapping iff $𝒫$ is a csf -network (resp. snf -network) of $X$ for a Ponomarev-system $(f,M,X,𝒫)$; (2) $f$ is a sequence-covering (resp. 1-sequence-covering) mapping iff every ${𝒫}_n$ is a cs-cover (resp. wsn-cover) of$X$ for a Ponomarev-system $(f,M,X,\{{𝒫}_n\})$. As applications of these results, some relations between sequence-covering mappings...