The Closed Graph and P-Ciosed Graph Properties in General Topology

T. R. Hamlett L. L. Herrington

AMERICAn MATHEMATICAL SOCIETY u·OLUME 3

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http://dx.doi.org/10.1090/conm/003

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The Closed Graph and P-Ciosed Graph Properties in General Topology

T. A. Hamlett L. L. Herrington

AMERICAn MATHEMATICAL SOCIETY PrOVIdence • Rlllda Island

Library of Congress Cataloging in Publication Data

Hamlett, T. R., 1950-The closed graph and P-dosed graph properties in general topology. (Contemporary mathematics; v. 3) Bibliography: p. 1. Topological spaces. 2. Closed graph theorems.

I. Herrington, L. L., 1944- . II. Title. III. Series: Contemporary mathematics (American Mathematical Society); v. 3. QA611.3.H35 514'.322 ISBN 0-8218-5004-0 ISSN 0271-4132

81-10888 AACR2

1980 Mathematics Subject Classification. Primary 54CIO; Secondary 54025, 54C05.

Copyright© 1981 by the American Mathcmatic.:al Society

Printed in the United States of America.

All rights reserved.

This book may not be reproduced in any form

without the permission of the publishers.

To Paul E. Long, Professor of Mathematics,

University of Arkansas

TABLE OF CONTENTS

Preface.

Abstract

PART I

1. Basic Definitions and Results.

2. The Closed Graph and Minimal Topological Spaces.

3. Some Applications to Functional Analysis

4. Non-continuous Functions and the Closed Graph Property

5. Characterizations of Compactness and Countable Compactness

6. Points of Discontinuity ....

PART II

l. Preliminary Definitions and Theorems ..

2. Properties of e-closed Graphs with Respect to Y.

3. H(i) Spaces and e-closed Graphs with Respect toY.

4. Characterizations of C-compact, H-closed, and Minimal Hausdorff Spaces . . . . . . . . . . . . . . . . .

5. Functions with a P-closed Graph with Respect to Y.

6. Functions with a P-closed Graph with Respect to X.

Bibliography .

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ix

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3

13

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19

25

29

35

37

41

47

51

57

65

PREFACE

In General Topology the basic setting (particularly from a categorical point of view) is a function F:X~Y where X and Y are arbitrary topological spaces. The fundamental consideration in this setting is: When is F con-tinuous? In the context of topological vector spaces, where F is linear and X and Y are Banach spaces, it is sufficient for continuity to show that F has a closed graph. Consequently, the Closed Graph Theorem has long been recognized as a major tool in Functional Analysis. Until recently, by comparison, very little effort has been expended in the setting of General Topology to determine when a function with a closed graph is continuous. In recent years, however, interest has grown in this topic simultaneously with an increasing interest in the more general subject of non-continuous functions.

Our purpose in writing this paper is to provide an elementary exposition of the results relating to functions with closed and P-closed graphs in a setting of General Topology. Some of the results included in this paper have not been published elsewhere, and represent the latest research efforts in this area.

While we have attempted to make the bibliography as complete as possible, the authors have not made a serious attempt to see that each theorem be assigned a reference in order to give credit to a particular mathematician.

Throughout the paper, X and Y denote arbitrary topological spaces with other hypotheses stated as needed. C1(A) denotes the closure of A, Int(A) the interior of A, N(x) denotes the neighborhood system at x, and nbd is used to abbreviate "neighborhood". CGT is occasionally used to abbreviate "Closed Graph Theorem".

Theorems, corollaries, major definitions, and examples are numbered by Part (roman numeral), Section (arabic numeral), and order of occurrence; for example, Theorem II.3.14 is Part II, Section 3, and there are 13 numbered items which precede it in Section 3.

It would be impossible to thank all the people to whom the authors are indebted for the writing of this paper. We want to mention, however, the names of Paul E. Long and W. T. Thompson, both of whom have made significant contri-butions to the research contained in this paper.

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ABSTRACT

PART I

1. BASIC DEFINITIONS AND RESULTS - Using the classical Closed Graph Theorem for Banach spaces as a starting point, several different characterizations of a function having a closed graph are provided. The concept of a "cluster set" of a function at a point is used to simplify the proofs of some well known theorems. Here some of the standard results are included, such as the ones in 12 and 43 .

2. THE CLOSED GRAPH AND MINIMAL TOPOLOGICAL SPACES - There are several gener-alizations of compact and compact Hausdorff spaces of current interest. These spaces are generally referred to as minimal topological spaces. In section I.l, compactness plays a very important role. In this section we show how the closed graph property interacts with these more general spaces.

3. SOME APPLICATIONS TO FUNCTIONAL ANALYSIS - The Closed Graph Theorem is a very powerful tool in Banach spaces. Our investigation of the closed graph property in the more general setting of functions in topological spaces of course leads to sufficient conditions for and characterizations of the closed graph property, which have implications in Banach spaces. In particular, the Closed Graph Theorem is rephrased in a new and interesting way.

4. NON-CONTINUOUS FUNCTIONS AND THE CLOSED GRAPH PROPERTY - Functions with closed graphs are but one of many types of non-continuous functions. In this section the relationship between the closed graph property and some of these functions is explored. This investigation leads to several results which give sufficient conditions for continuity.

5. CHARACTERIZATIONS OF COMPACTNESS AND COUNTABLE COMPACTNESS - Th~re is an interesting relationship between the concepts of compactness and the closed graph property. It is shown how the closed graph property can be used to characterize compactness and countable compactness where the space in question is used as a domain or a range space.

6. POINTS OF DISCONTINUITY - Since functions that have closed graphs and con-tinuous functions have many similar properties, the set of discontinuities of a function with a closed graph should, and does, lend itself to investigation. The Baire space property, perfect normality, and metric spaces with the property that bounded sets have compact closure, all play an important role in describing this set as closed and nowhere dense under appropriate conditions.

X

ABSTRACT

PART II

1. PRELIMINARY DEFINITIONS AND THEOREMS - The concepts of 0-convergence and 0-accumulation in terms of filterbases and nets are defined and discussed.

2. PROPERTIES OF e-CLOSED GRAPHS WITH RESPECT TO Y - The convergence concepts introduced in section II.l are used to define thee-closed graph with respect toY (w.r.t. Y) property. Characterizations are given and relationships with certain other non-continuous functions are discussed.

3. H(i} SPACES AND e-CLOSED GRAPHS WITH RESPECT TO Y - Functions with a e-closed graph w.r.t. Y are shown to be natural companions for H(i) spaces. T1 H(i} spaces are characterized and sufficient conditions for continuity are investigated.

4. CHARACTERIZATIONS OF C-COMPACT, H-CLOSED, AND MINIMAL HAUSDORFF SPACES -These spaces are characterized in terms of functions with a e-closed graph w.r.t. Y.

5. FUNCTIONS WITH A P-CLOSED GRAPH WITH RESPECT TO Y - Urysohn-closed, com-pletely Hausdorff-closed, and regular-closed spaces are characterized using appropriate types of closed graph properties of functions.

6. FUNCTIONS WITH A P-CLOSED GRAPH WITH RESPECT TO X - H-closed, functionally compact, and C-compact spaces are characterized in terms of functions with a e-closed graph w.r.t. X. Completely Hausdorff-closed spaces are characterized in terms of functions with an f-closed graph w.r.t. X.

• ..

xi