Continuous Convergence on C(X)
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469 Ernst Binz
Continuous Convergence on C(X)
Springer-Verlag 9 York 19 75 Berlin-Heidelberg New
Author Prof. Dr. Ernst Binz Universit~t Mannheim (WH) Lehrstuhl fLir Mathematik I 68 Mannheim SchloB BRD
Library of Coagress Cataloging in Publication Data
Binz, Ernst, 1939Continuous convergence on C(X) (Lecture notes in mathematics ; 469) Bibliography: p. lneludes index. i. Function spaces. 2. Convergence. 3. Topological algebras. I. Title. If. Series : Lecture notes in mathematics (Berlin) ; 469. QA3.L28 no. ~69 [QA523] 510'8s [532' .55] 75-16495
AMS Subject Classifications (1970): 22-02, 22A99, 46-02, 46E99, 46 H 99, 46 M 99, 54-02, 54 A 20, 54 C35, 54 C 99, 54 H 10 ISBN 3-540-07179-2 Springer-Verlag Berlin Heidelberg 9 New 9 York 1SBN 0-387-07179-2 Springer-Verlag New York Heidelberg 9 Berlin 9 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin Heidelberg 9 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
T
a
:
Erika~
Barbara
and
Dieter
ACKNOWLEDGEMENT
Much of the material presented
in these notes was worked out in colla-
boration with K.Kutzler and my former students: man and M.Schroder.
W.A.Feld-
I am greatly indebted to all of them.
Special thanks go to H.P.Butzman suggestions
H.P.Butzmann,
and B.MGller
for their criticisms,
and the painful job of proofreading.
The suggestion
of C.H.Cook to write these notes in English provided
him and Dany Gulick with some work of linguistic
adjustments.
I am
grateful to both of them.
The manuscript
was typed by Mms.K.Bischoff.
I would
very much for the care she took with this job.
like to thank her
TABLE
OF C O N T E N T S
INTRODUCTION ....................................................
VII
O.
i
CONVERGENCE
SPACES ...........................................
0.1
Convergence
0.2
The
1. F U N C T I O N I.i
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
structure
of c o n t i n u o u s
completely
a convergence
2. V E C T O R
convergence ................
ALGEBRAS ............................................
The
1.2
i
regular
topological
associated
7 to
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Realcompactification
and
a completely
topological
SPACE
space
regular
TOPOLOGIES
ON C(X)
Stone-Cech
compactification
7 of
space . . . . . . . . . . . . . . . . .
FOR WHICH
THE EVALUATION
5
I0
MAP
IS C O N T I N U O U S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1
A natural
on C(X) . . . . . . . . . . . . . .
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