Generalized Metric Spaces and Mappings
The idea of mutual classification of spaces and mappings is one of the main research directions of point set topology. In a systematical way, this book discusses the basic theory of generalized metric spaces by using the mapping method, and summarizes the
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Shou Lin Ziqiu Yun
Generalized Metric Spaces and Mappings
Atlantis Studies in Mathematics Volume 6
Series editor Jan van Mill, VU University, Amsterdam, The Netherlands
Aims and Scope With this book series, we aim to publish monographs of high quality in all areas of mathematics. Both research monographs and books of an expository nature are welcome. This series is the continuation of the “Mathematics Studies”, previously published by Elsevier. All books published after November 2010 are promoted, distributed and sold by Springer, both as e-books and in print. The books are also part of SpringerLink and included in the relevant Springer subject collections. All book proposals submitted to this series will be reviewed by the Series Editor. After the manuscript has been completed, it will be entirely reviewed by one of our editors or reviewers. Only after this review will the book be published.
More information about this series at http://www.springer.com/series/10070
Shou Lin Ziqiu Yun •
Generalized Metric Spaces and Mappings
Shou Lin School of Mathematics and Statistics Minnan Normal University Zhangzhou China
ISSN 1875-7634 Atlantis Studies in Mathematics ISBN 978-94-6239-215-1 DOI 10.2991/978-94-6239-216-8
Ziqiu Yun Department of Mathematics Soochow University Suzhou China
ISSN 2215-1885
(electronic)
ISBN 978-94-6239-216-8
(eBook)
Library of Congress Control Number: 2016951688 Translation and revision from the Chinese language edition: 广义度量空间与映射 by Shou Lin © Science Press, Beijing 2007. All Rights Reserved. © Atlantis Press and the author(s) 2016 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
Foreword
What are generalized metric spaces? Most often, this expression denotes a system of classes of topological spaces, each of which is defined with the help of some typical topological property of metrizable spaces. For example, the classes of paracompact spaces, Moore spaces, spaces with a point-countable base, submetrizable spaces, perfectly normal spaces, first-countable spaces belong to this system. We also include in it the class of symmetrizable spaces and the class of Dmetrizable spaces. The topologies of these spaces are generated by generalized metrics. In this way, we obtain a rich system of classes of topological spaces which are all emerging, growing and spreading in many directions, from the same powerful germ—the concept of the topology generated by a metric. The system of generalized metrizable spaces (below, we often use an abbreviation “GMS-system” to denote it) also includes various relations between its members. The simplest among them is, of course, the inclusion relation. But much deeper, often unpredictable links between classes of topological spaces are established by means of mappings. Mappings of various kinds se
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