Volume 61 , issue 3 ( 2009 )back
A new hyperspace topology and the study of the function space $\theta^*$-$LC(X,Y)$181$-$193
S. Ganguly, Sandip Jana and Ritu Sen

Abstract

The intent of this paper is to introduce a new hyperspace topology on the collection of all $\theta$-closed subsets of a topological space. The space of all $\theta^*$-lower semicontinuous functions has been studied in detail and finally we deal with some multifunctions.

Creative Commons License

Keywords: $\theta$-closed set; $H$-closed space; $H$-set; $\theta$-partially ordered space; $\theta^*$-lower semicontinuous functions; multifunctions.

MSC: 54B20, 54C35

Selection principles and Baire spaces195$-$202
Marion Scheepers

Abstract

We prove that if $X$ is a separable metric space with the Hurewicz covering property, then the Banach-Mazur game played on $X$ is determined. The implication is not true when ``Hurewicz covering property" is replaced with ``Menger covering property".

Creative Commons License

Keywords: Baire space; First category; Banach-Mazur game; Menger property; Hurewicz property.

MSC: 03E99, 54D20, 54E52

Some results in fixed point theory concerning generalized metric spaces203$-$208
Ali Fora, Azzeddine Bellour and Adnan Al-Bsoul

Abstract

In this paper we shall study the fixed point theory in generalized metric spaces (gms). One of our results will be a generalization of Kannan's fixed point theorem in the ordinary metric spaces, and Das's fixed point theorem in gms.

Creative Commons License

Keywords: Generalized metric space; Fixed point.

MSC: 54H25, 47H10

On $so$-metrizable spaces209$-$218
Xun Ge

Abstract

In this paper, we give some new characterizations for $so$-metrizable spaces, which answers a question posed by Z. Li and generalize some results on $so$-metrizable spaces. As some applications of the above results, some mappings theorems on $so$-metrizable spaces are obtained.

Creative Commons License

Keywords: $so$-network,; $sof$-countable; $so$-metrizable space.

MSC: 54C10, 54D50, 54E35, 54E99

On $L^{1}$-convergence of certain generalized modified trigonometric sums219$-$226
Karanvir Singh and Kulwinder Kaur

Abstract

In this paper we define new modified generalized sine sums $K_{nr}(x)=\dfrac{1}{2\sin x}\sum_{k=1}^{n}(\triangle^{r}a_{k-1}-\triangle^{r}a_{k+1}) \tilde{S}_{k}^{r-1}(x)$ and study their $L^{1}$-convergence under a newly defined class $\bold{K}^{\alpha}$. Our results generalize the corresponding results of Kaur, Bhatia and Ram [6] and Kaur~[7].

Creative Commons License

Keywords: $L^{1}-$convergence; conjugate Ces\`{a}ro means; generalized sine sums.

MSC: 42A20, 42A32

Compactness and weak compactness of elementary operators on $B(l^2)$ induced by composition operators on $l^2$227$-$233
Gyan Prakash Tripathi

Abstract

In this paper we have given simple proofs of some range inclusion results of elementary operators on $B(l^{2})$ induced by composition operators on $l^{2}$. By using these results we have characterized compact and weakly compact elementary operators on $B(l^{2})$ induced by composition operators on $l^{2}$.

Creative Commons License

Keywords: Compactness; composition operators; elementary operators; thin operators.

MSC: 47B33, 47B47

Riesz spaces of measures on semirings235$-$239
Z. Ercan

Abstract

It is shown that the spaces of finite valued signed measures (signed charges) on $\sigma$-semirings (semirings) are Dedekind complete Riesz spaces, which generalizes known results on $\sigma$-algebra and algebra cases.

Creative Commons License

Keywords: Riesz spaces; semiring; measure

MSC: 28C99, 46G12

Characterizations of $\delta$-stratifiable spaces241$-$246
Kedian Li

Abstract

In this paper, we give some characterizations of $\delta$-stratifiable spaces by means of $g$-functions and semi-continuous functions. It is established that: ıtem{(1)} A topological space $X$ in which every point is a regular $G_\delta$-set is $\delta$-stratifiable if and only if there exists a $g$-function $g:N\times X\rightarrow \tau $ satisfies that if $Fın RG(X)$ and $y\notin F$, then there is an $mın N$ such that $y\notin \overline{g(m,F)}$; ıtem{(2)} If there is an order preserving map $\varphi:USC(X)\rightarrow LSC(X) $ such that for any $hın USC(X),0\leq \varphi(h)\leq h$ and $0<\varphi(h)(x)0$, then $X$ is $\delta$-stratifiable space.

Creative Commons License

Keywords: $\delta$-stratifiable spaces; $g$-functions; upper semi-continuous maps; lower semi-continuous maps.

MSC: 54E20, 54C08; 26A15