Volume 61 , issue 3 ( 2009 ) | back |

A new hyperspace topology and the study of the function space $\theta^*$-$LC(X,Y)$ | 181$-$193 |

**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.

**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 spaces | 195$-$202 |

**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".

**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 spaces | 203$-$208 |

**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.

**Keywords:** Generalized metric space; Fixed point.

**MSC:** 54H25, 47H10

On $so$-metrizable spaces | 209$-$218 |

**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.

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

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

On $L^{1}$-convergence of certain generalized modified trigonometric sums | 219$-$226 |

**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].

**Keywords:** $L^{1}-$convergence; conjugate Cesàro 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 |

**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 $l^{2}$.

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

**MSC:** 47B33, 47B47

Riesz spaces of measures on semirings | 235$-$239 |

**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.

**Keywords:** Riesz spaces; semiring; measure

**MSC:** 28C99, 46G12

Characterizations of $\delta$-stratifiable spaces | 241$-$246 |

**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:
\item{(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 $F\in RG(X)$ and $y\notin F$, then there is an $m\in N$
such that $y\notin \overline{g(m,F)}$;
\item{(2)} If there is an order preserving map $\varphi:USC(X)\rightarrow LSC(X) $ such
that for any $h\in USC(X),0\leq \varphi(h)\leq h$ and
$0<\varphi(h)(x)

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

**MSC:** 54E20, 54C08; 26A15