Let $\left\{{𝒫}_n\right\}$ be a sequence of covers of a space $X$ such that $\left\{st(x,{𝒫}_n)\right\}$ is a network at $x$ in $X$ for each $x\in X$. For each $n\in \mathbb{N}$, let ${𝒫}_n=\{P_{\beta }:\beta \in {\Lambda }_n\}$ and ${\Lambda }_n$ be endowed the discrete topology. Put $M=\{b=\left({\beta }_n\right)\in {\Pi }_{n\in \mathbb{N}}{\Lambda }_n:\left\{P_{{\beta }_n}\right\}$ forms a network at some point $x_{b}inX\}$ and $f:M⟶X$ by choosing $f\left(b\right)=x_b$ for each $b\in M$. In this paper, we prove that $f$ is a sequentially-quotient (resp. sequence-covering, compact-covering) mapping if and only if each ${𝒫}_n$ is a $c{s}^{*}$-cover (resp. $fcs$-cover, $cfp$-cover) of $X$. As a consequence of this result, we prove that $f$ is a sequentially-quotient, $s$-mapping if and...

It is shown that every strong $\Sigma$ space is a $D$-space. In particular, it follows that every paracompact $\Sigma$ space is a $D$-space.

Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D,p)$, and also prove that if the equation $U^2-D{V}^2=-1$ has integer solutions $(U,V)$, the least solution $(u_1,v_1)$ of the equation $u^2-p{v}^2=1$ satisfies $p\nmid v_1$, and $D>C\left(p\right)$, where $C\left(p\right)$ is an effectively computable...