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Abstract functions with continuous differences and Namioka spaces

Authors: Bolis Basit and Hans Günzler
Journal: Trans. Amer. Math. Soc. 348 (1996), 4489-4500
MSC (1991): Primary 28B05, 39A05; Secondary 90D05, 54C05, 54E35
MathSciNet review: 1373629
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Abstract: Let $G$ be a semigroup and a topological space. Let $X$ be an Abelian topological group. The right differences $\triangle _{h} \varphi $ of a function $\varphi : G \to X$ are defined by $\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)$ for $h,t \in G$. Let $\triangle _{h} \varphi $ be continuous at the identity $e$ of $G$ for all $h$ in a neighbourhood $U$ of $e$. We give conditions on $X$ or range $\varphi $ under which $\varphi $ is continuous for any topological space $G$. We also seek conditions on $G$ under which we conclude that $\varphi $ is continuous at $e$ for arbitrary $X$. This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

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Additional Information

Bolis Basit
Affiliation: Department of Mathematics, Monash University, Clayton Vic. 3168, Australia

Hans Günzler
Affiliation: Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str., 424098 Kiel, Deutschland

Keywords: Differences, weak continuity, Namioka spaces, right uniform continuity, Baire spaces, Banach spaces not containing $c_{0}$
Received by editor(s): May 4, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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