Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_{\delta }$-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\left\{x\right\}=\bigcap \left\{g\right(x,n):n\in \mathbb{N}\}$ and $g(x,n+1)\subseteq g(x,n)$ for each...

We study the solutions and attractivity of the difference equation $x_{n+1}=x_{n-3}/(-1+x_n{x}_{n-1}{x}_{n-2}{x}_{n-3})$ for $n=0,1,2,\cdots$ where $x_{-3},x_{-2},x_{-1}$ and $x_0$ are real numbers such that $x_0{x}_{-1}{x}_{-2}{x}_{-3}\ne 1.$

A sequence of random elements $\{X_j,j\in J\}$ is called strongly tight if for an arbitrary $ϵ>0$ there exists a compact set $K$ such that $P\left({\bigcap }_{j\in J}[X_j\in K]\right)>1-ϵ$. For the Polish space valued sequences of random elements we show that almost sure convergence of $\left\{X_n\right\}$ as well as weak convergence of randomly indexed sequence $\left\{X_{\tau }\right\}$ assure strong tightness of $\{X_n,n\in \mathbb{N}\}$. For $L^1$ bounded Banach space valued asymptotic martingales strong tightness also turns out to the sufficient condition of convergence. A sequence of r.e. $\{X_n,n\in \mathbb{N}\}$ is said to converge essentially...