Book:Steven A. Gaal/Point Set Topology

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


 * AUTHOR INDEX
 * SUBJECT INDEX



Source work progress
* : $\S 1.1$: Exercise $1 \ \text{(c)}$