, INTRODUCTORY
TOPOLOGY
Exercises and Solutions
Second Edition
Mohammed Hichem Mortad
University of Oran 1, Ahmed Ben Bella, Algeria
World Scientific
NEW JERSEY • LONDON • SI NGAP ORE • BEIJING • S HAN G HAI • HONG KONG • TAIPEI • CHENNAI
,Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Mortad, Mohammed Hichem, 1978-
Title: Introductory topology : exercises and solutions / by Mohammed Hichem Mortad
(University o f Oran, Algeria).
Description: 2nd edition. |New Jersey : World Scientific, 2016. | Includes
bibliographical references and index.
Identifiers: LCCN 2016030117 |ISBN 9789813146938 (hardcover : alk. paper) |
ISBN 9789813148024 (pbk : alk. paper)
Subjects: LCSH: Topology—Problems, exercises, etc.
Classification: LCC QA611 .M677 2016 |DDC 514.076«dc23
LC record available at https://lccn.loc.gov/2016030117
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
A ll rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
be invented, without written permission from the Publisher.
For photocopying o f material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not
required from the publisher.
Printed in Singapore
, Contents
Preface ix
Preface to the Second Edition xiii
Notation and Terminology XV
0.1. Notation X V
0.2. Terminology xvi
Part 1. Exercises 1
Chapter 1. General Notions: Sets, Functions et al. 3
1 .1 . Essential Background 3
1 .1 .1 . Sets 3
1.1.2. Functions 9
1.1.3. Equivalence Relation 15
1.1.4. Consequences of The Least Upper Bound Property 15
1.1.5. Countability 17
1.2. Exercises With Solutions 19
1.3. More Exercises 24
Chapter 2. Metric Spaces 27
2.1. Essential Background 27
2.1.1. Definitions and Examples 27
2.1.2. Important Sets in Metric Spaces 30
2.1.3. Continuity in Metric Spaces 32
2.1.4. Equivalent Metrics 34
2.2. True or False: Questions 36
2.3. Exercises With Solutions 36
2.4. Tests 42
2.5. More Exercises 43
Chapter 3. Topological Spaces 47
3.1. Essential Background 47
3.1.1. General Notions 47
3.1.2. Separation Axioms 49
3.1.3. Closures, Interiors, Limits Points, et al 50
v
TOPOLOGY
Exercises and Solutions
Second Edition
Mohammed Hichem Mortad
University of Oran 1, Ahmed Ben Bella, Algeria
World Scientific
NEW JERSEY • LONDON • SI NGAP ORE • BEIJING • S HAN G HAI • HONG KONG • TAIPEI • CHENNAI
,Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Mortad, Mohammed Hichem, 1978-
Title: Introductory topology : exercises and solutions / by Mohammed Hichem Mortad
(University o f Oran, Algeria).
Description: 2nd edition. |New Jersey : World Scientific, 2016. | Includes
bibliographical references and index.
Identifiers: LCCN 2016030117 |ISBN 9789813146938 (hardcover : alk. paper) |
ISBN 9789813148024 (pbk : alk. paper)
Subjects: LCSH: Topology—Problems, exercises, etc.
Classification: LCC QA611 .M677 2016 |DDC 514.076«dc23
LC record available at https://lccn.loc.gov/2016030117
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
A ll rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
be invented, without written permission from the Publisher.
For photocopying o f material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not
required from the publisher.
Printed in Singapore
, Contents
Preface ix
Preface to the Second Edition xiii
Notation and Terminology XV
0.1. Notation X V
0.2. Terminology xvi
Part 1. Exercises 1
Chapter 1. General Notions: Sets, Functions et al. 3
1 .1 . Essential Background 3
1 .1 .1 . Sets 3
1.1.2. Functions 9
1.1.3. Equivalence Relation 15
1.1.4. Consequences of The Least Upper Bound Property 15
1.1.5. Countability 17
1.2. Exercises With Solutions 19
1.3. More Exercises 24
Chapter 2. Metric Spaces 27
2.1. Essential Background 27
2.1.1. Definitions and Examples 27
2.1.2. Important Sets in Metric Spaces 30
2.1.3. Continuity in Metric Spaces 32
2.1.4. Equivalent Metrics 34
2.2. True or False: Questions 36
2.3. Exercises With Solutions 36
2.4. Tests 42
2.5. More Exercises 43
Chapter 3. Topological Spaces 47
3.1. Essential Background 47
3.1.1. General Notions 47
3.1.2. Separation Axioms 49
3.1.3. Closures, Interiors, Limits Points, et al 50
v
