Abstract

In this paper we study the characterizations of compact-covering and 1-sequence-covering (resp. 2-sequence-covering) images of metric spaces and give a positive answer to the following question: How to characterize first countable spaces whose each compact subset is metrizable by certain images of metric spaces?

Keywords: Metrizable spaces; compact-covering maps; 1-sequence-covering maps; $so$-network; $snf$-countable spaces.

MSC: 54C10, 54D70, 54E40, 54E20, 54E99

Abstract

Let $I$ be a nonzero ideal of an integral domain $T$ and let $\varphi\:T\to T/I$ be the canonical surjection. If $D$ is an integral domain contained in $T/I$, then $R=\varphi^{-1}\left(D\right)$ arises as a pullback of type $\square$ in the sense of Houston and Taylor such that $R\subseteq T$ is a domains extension. The stability of atomic domains, domains satisfying ACCP, HFDs, valuation domains, PVDs, AVDs, APVDs and PAVDs observed on all corners of pullback of type $\square$ under the assumption that the domain extension $R\subseteq T$ satisfies $Condition$ $1:$ For each $b\in T$ there exist $u\in\cup(T)$ and $a\in R$ such that $b=ua$.

Keywords: Pullback; condition 1.

MSC: 13G05, 16U10

Abstract

Write $c$ for the cardinality of the continuum and let $\eta$ be the Euclidean topology on ${\Bbb R}$. Let $\Sigma$ be the family of all $\sigma$-ideals ${\cal I}$ on ${\Bbb R}$ such that $\bigcup{\cal I}$ is dense and ${\Bbb Q}\cap\bigcup{\cal I}=\emptyset$. Then for each ${\cal I}\in\Sigma$ the family $\eta/{\cal I}$ of all sets $X\setminus Y$ with $X\in\eta$ and $Y\in{\cal I}$ is a topology on ${\Bbb R}$. Such a refinement of $\eta$ always preserves separability and connectedness, but destroys metrizability (and first countability almost always) and makes the space totally pathwise disconnected. Nevertheless, the separable Hausdorff space $({\Bbb R},\eta/{\cal I})$ still has the two metric properties that every point is reachable by a sequence of points within any fixed countable dense set and that (even in the absence of first countability) sequential continuity is strong enough to entail continuity. In detail we investigate further main properties in the four most interesting cases when the $\sigma$-ideal ${\cal I}$ consists of either all countable sets or all null sets or all meager sets or all sets contained in ${\Bbb R}\setminus{\Bbb Q}$. Finally we track down a subfamily $\Sigma_1$ of $\Sigma$ with cardinality $2^{2^c}$ such that $({\Bbb R},\eta/{\cal I})$ and $({\Bbb R},\eta/{\cal J})$ are never homeomorphic for distinct ${\cal I},{\cal J}$ in $\Sigma_1$.

Keywords: $\sigma$-ideal; Lebesgue null set; meager; separable; totally pathwise disconnected.

MSC: 54G20, 54A10

Abstract

We prove a coupled random coincidence and a coupled random fixed point theorem under a set of conditions. Our result is a generalization of the recent result of Ćirić and Lakshmikantham [Stochastic Analysis and Applications 27:6 (2009), 1246--1259].

Keywords: Common fixed point; coupled coincidence fixed point; measurable mapping; random operator.

MSC: 54H25, 47H10, 54E50

Abstract

This paper deals with the Jacobi type and Gegenbauer type generalizations of certain polynomials and their generating functions. Relationships among those generalized polynomials have also been indicated.

Keywords: Jacobi type and Gegenbauer type generalization of Sister Celine's polynomials, Bateman's polynomials, Pasternack's polynomials, Hahn polynomial, Rice polynomials, generating functions.

MSC: 33C45

Abstract

A space $X$ is {\it almost countably compact\/} if for every countable open cover $\cal U$ of $X$, there exists a finite subset $\cal V$ of $\cal U$ such that $\bigcup\{\overline{V}:V\in \cal V\}=X$. In this paper, we investigate the relationship between almost countably compact spaces and countably compact spaces, and also study topological properties of almost countably compact spaces.

Keywords: Compact; countably compact; almost countably compact.

MSC: 54D20, 54D55

Abstract

New sufficient conditions are derived for the integral operator of mereomorphic functions defined by $$ H(z)=\frac{c}{z^{c+1}}\int_{0}^{z}u^{c-1}(uf_{1}(u))^{\gamma_{1}}\dotsm(uf_{n}(u))^{\gamma _{n}}\,du, $$ to be in the class $\Sigma_{N}(\lambda)$ of meromorphic functions satisfying the condition $-\Re\{\frac{zf^{\prime \prime}(z)}{f^{\prime}(z)}+1\}<\lambda$, where $\lambda>1$.

Keywords: Meromorphic functions; integral operator

MSC: 30C45