Invariants for Real-Generated Uniform Topological and Algebraic Categories
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491 Kevin A. Broughan
Invariants for Real-Generated Uniform Topological and Algebraic Categories
Springer-Verlag Berlin.Heidelberg- New York 1975
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
491 Kevin A. Broughan
Invariants for Real-Generated Uniform Topological and Algebraic Categories
Springer-Verlag Berlin.Heidelberg- New York 1975
Author Kevin A. Broughan Department of Mathematics University of Waikato Hamilton/New Zealand
Library of Congress Cataloging in Publication Data
Broughan, K ~ 1943lnvariant for real-generated braic categories.
u n i f o r m and a l g e -
( L e c t u r e n o t e s i n m a t h e m a t i c s ; 491) Bibliography: p. Includes index. 1. tLniform s p a c e s . 2. T o p o l o g i c a l s p a c e s . 3o Categories (Mathematics) 4. Invariants. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 491. QA3.L28 no. 491 [QA611.25] 510'.8s [514'.32] 75-34130
AMS Subject Classifications (1970): 02E15, 02E99, 06A10, 10A40, 10B05, 10F35, 10M10, 12J10, 12J20, 13A15,13J99,15A03, 18B99, 18D35,22A99, 26A51, 33A70, 46A15, 54C05, 54C30, 54D20, 54 E05, 54E15, 54 E25, 54 E35, 54 F45, 54 F50, 54 H25. ISBN 3-540-07418-X Springer-Verlag Berlin Heidelberg 9 New 9 York ISBN 0-387-07418-X 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 9Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Introduction
This natural
book
is c o n c e r n e d w i t h
invariants
for c a t e g o r i e s
of u n i f o r m and t o p o l o g i c a l categories tion and
the n a t u r a l
invariants
of c o n s t r u c t i o n
spaces.
metric having numbers,
are
similar
of the
corresponding
from the
invariants
in the c a t e g o r y
If a t o p o l o g i c a l its range
metric.
real n u m b e r s
metrization
which
However,
give rise
to u n i q u e m e t r i z a t i o n
theorems.
give rise
to the same m e t r i z a t i o n More precisely,
metrizable
topological
its r a n g e
are
topolo-
a compatible
such a
s u b s e t of the p o s i t i v e
chosen.
real n u m b e r s
-- this
s u b s e t of the real
of the
to be e q u i v a l e n t .
subset
follows
at least t h e o r e t i c a l l y ,
theorem,
the
of the
space has also
for e a c h
t h e r e w i l l be,
generated
space has
in a c e r t a i n
Hence,
set of
of m e t r i z a b l e
structure
having
in d e f i n i -
An i l l u s t r a t i o n
then e v e r y h o m e o m o r p h i c
compatible
positive
categories
In each of t h e s e
functions which have
in each category.
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