Book:Steven A. Gaal/Point Set Topology

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Steven A. Gaal: Point Set Topology

Published $1964$, Dover

ISBN 0-486-47222-1.


Subject Matter


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
AUTHOR INDEX
SUBJECT INDEX