Spaces on Which Every Pointwise Convergent Series of Continuous Functions Converges Pseudo-normally
نویسنده
چکیده
A topological space X is a ΣΣ∗-space provided for every sequence 〈fn〉n=0 of continuous functions from X to R, if the series ∑∞ n=0 |fn| converges pointwise then it converges pseudo-normally. We show that every regular Lindelöf ΣΣ∗-space has Rothberger property. We also construct, under the continuum hypothesis, a ΣΣ∗-subset of R of cardinality continuum.
منابع مشابه
POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS
We study the space of all continuous fuzzy-valued functions from a space $X$ into the space of fuzzy numbers $(mathbb{E}sp{1},dsb{infty})$ endowed with the pointwise convergence topology. Our results generalize the classical ones for continuous real-valued functions. The field of applications of this approach seems to be large, since the classical case allows many known devices to be fi...
متن کاملUnconditional Convergence in Banach Spaces
Introduction. This note investigates an apparent generalization of unconditionally convergent series ^ x » in weakly complete Banach spaces. A series of elements with Xi in E is said to be unconditionally convergent if for every variation of sign €j= ± 1 , ^TMeiXi is convergent. This formulation of the definition of unconditional convergence is equivalent to that given by Orliczjé]. We call ^Xi...
متن کاملOperator Valued Series and Vector Valued Multiplier Spaces
Let $X,Y$ be normed spaces with $L(X,Y)$ the space of continuous linear operators from $X$ into $Y$. If ${T_{j}}$ is a sequence in $L(X,Y)$, the (bounded) multiplier space for the series $sum T_{j}$ is defined to be [ M^{infty}(sum T_{j})={{x_{j}}in l^{infty}(X):sum_{j=1}^{infty}% T_{j}x_{j}text{ }converges} ] and the summing operator $S:M^{infty}(sum T_{j})rightarrow Y$ associat...
متن کاملConvergence of Automorphisms of Compact Projective Planes
We show that a pointwise convergent sequence (n) n2IN of continuous collineations of a compact projective plane converges uniformly if and only if the pointwise limit of (n) n2IN has a quadrangle in its image. Moreover is then a continuous collineation. Furthermore, we derive that a homomorphism between topological projective planes is continuous if and only if it is continuous at some point.
متن کاملRepresentability of Certain Function Spaces
LetCp(X) be the space of all continuous real-valued functions on a space X , with the topology of pointwise convergence. In this paper we show thatCp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain r...
متن کامل