ProofWiki:Books/Steven A. Gaal/Point Set Topology

Steven A. Gaal: Point Set Topology

Published 1964, Dover.

ISBN 0-486-47222-1

Subject Matter

• Topology

Contents

PREFACE

NOTATION

Introduction to Set Theory

1. ELEMENTARY OPERATIONS ON SETS

2. SET THEORETICAL EQUIVALENCE AND DENUMERABILITY

3. THE AXIOM OF CHOICE AND ITS EQUIVALENTS

NOTES

REFERENCES

Chapter I Topological Spaces

1. OPEN SETS AND CLOSED SETS

2. INTERIOR, EXTERIOR, BOUNDARY, AND CLOSURE

3. CLOSURE OPERATORS

4. BASES AND SUBBASES

5. TOPOLOGIES ON LINEARLY ORDERED SETS

6. METRIC SPACES

7. NEIGHBORHOOD FILTERS

8. UNIFORM STRUCTURES

9. SIMPLE RESULTS ON UNIFORM STRUCTURES AND UNIFORM SPACES

10. SUBSPACES

11. PRODUCT SPACES

12. PRODUCTS OF UNIFORMIZABLE SPACES

13. INVERSE AND DIRECT IMAGES OF TOPOLOGIES

14. QUOTIENT SPACES

NOTES

REFERENCES

Chapter II Separation Properties

1. $(T_0)$ AND $(T_1)$ AXIOMS, HAUSDORFF SPACES

2. $(T_3)$ SPACES, REGULAR AND SEMIREGULAR SPACES

3. $(T_4)$ SPACES AND NORMAL SPACES

4. POINT-FINITE AND STAR-FINITE OPEN COVERINGS

5. $(T_5)$ SPACES AND COMPLETELY NORMAL SPACES

6. SEPARATED SETS

7. CONNECTED SPACES AND SETS

8. MAXIMAL CONNECTED SUBSETS

9. $(T)$ AXIOM AND COMPLETE REGULARITY

10. UNIFORMIZATION AND AXIOM $(T)$

11. AXIOMS OF SEPARATION IN PRODUCT SPACES

12. SEPARABLE SPACES AND COUNTABILITY AXIOMS

NOTES

REFERENCES

Chapter III Compactness and Uniformization

1. COMPACTNESS

2. COMPACT METRIC SPACES

3. SUBSPACES AND SEPARATION PROPERTIES OF COMPACT SPACES

4. THE PRODUCT OF COMPACT TOPOLOGICAL SPACES

5. LOCALLY COMPACT SPACES

6. PARACOMPACTNESS AND FULL-NORMALITY

7. THE EQUIVALENCE OF PARACOMPACTNESS AND FULL-NORMALITY

8. METRIZABLE UNIFORM STRUCTURES AND STRUCTURE GAGES

9. METRIZABILITY CONDITIONS

NOTES

REFERENCES

Chapter IV Continuity

1. FUNCTIONAL RELATIONS AND FUNCTIONS

2. LOCAL CONTINUITY

3. CONTINUOUS FUNCTIONS

4. HOMEOMORPHISMS, OPEN AND CLOSED MAPS

5. REAL-VALUED FUNCTIONS

6. CONTINUITY AND AXIOMS OF SEPARATION

7. CONTINUITY AND COMPACTNESS

8. CONTINUITY AND CONNECTEDNESS

9. CONTINUITY IN PRODUCT SPACES

10. UNIFORM CONTINUITY AND EQUICONTINUITY

11. THE TOPOLOGY OF UNIFORM CONVERGENCE

12. THE ALGERBA OF CONTINUOUS FUNCTIONS

NOTES

REFERENCES

Chapter V Theory of Convergence

1. FILTERS AND NETS

2. CONVERGENCE OF FILTERS, NETS, AND SEQUENCES

3. ULTRAFILTERS AND UNIVERSAL NETS

4. BOUNDS, TRACES, AND PRODUCTS OF FILTERS

5. APPLICATIONS OF FILTERS AND NETS TO COMPACTNESS

6. CAUCHY FILTERS AND COMPLETE SPACES

7. COMPLETION OF METRIC STRUCTURES

8. BAIRE'S CATEGORY THEOREM, THE PRINCIPLES OF UNIFORM BOUNDEDNESS AND OF THE CONDENSATION OF SINGULARITIES

9. COMPLETIONS AND COMPACTIFICATIONS

NOTES

REFERENCES

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