General Topology II

§1. Compactness and Its Different Forms: Separation Axioms.- 1.1. Different Definitions of Compactness.- 1.2. Relative Compactness.- 1.3. Countable Compactness.- 1.4. Relative Countable Compactness.- 1.5. Pseudocompact Spaces.- 1.6. Separation Axioms and Properties Related to Compactness.- 1.7. Star Characterizations of Countable Compactness and Pseudocompactness.- §2. Compactness and Products.- 2.1. Tikhonov's Theorem on Compactness of the Product.- 2.2. Products of Countably Compact Spaces.- 2.3. Products of Pseudocompact Spaces.- 2.4. Total Countable Compactness and Total Pseudocompactness.- 2.5. Compactness with Respect to a Fixed Ultrafilter (?-Compactness).- 2.6. ?-Products of Compact Spaces.- §3. Continuous Mappings of Compact Spaces.- 3.1. Theorem on Compactness of the Image and Its Consequences.- 3.2. Continuous Images of "Standard" Compacta.- 3.3. Open Mappings of Compacta and Dimension.- 3.4. Mardesi?'s Factorization Theorem.- 3.5. Continuous Images of Ordered Compacta.- 3.6. Pseudocompactness and Continuous Mappings.- 3.7. Continuous Mappings and Extremally Disconnected Compacta.- 3.8. Scattered Compacta and Their Images.- §4. Metrizability Conditions for Compact, Countably Compact and Pseudocompact Spaces.- 4.1. Classical Results and the Theorem of Chaber.- 4.2. Theorems of Dow and Tkachenko.- 4.3. Point-countable and ?-Point-finite Bases.- 4.4. Quasi-developments and ??-Bases.- 4.5. Strongly N0-Noetherian Bases.- 4.6. Rank of a Base and Metrizability Conditions for Compacta.- 4.7. Symmetrics and Metrizability of Compacta.- §5. Cardinal Invariants in the Class of Compacta.- 5.1. Network Weight, Diagonal Number and Weight of Compacta.- 5.2. Pseudocharacter and Character in the Class of Compacta.- 5.3. First Countable Compacta.- 5.4. Perfectly Normal Compacta.- 5.5. Continuous Images of First Countable Compacta.- 5.6. Sequential Compacta and the First Axiom of Countability Almost Everywhere.- 5.7. Corson Compact Spaces and N0-Monolithicity.- 5.8. Compactaof Countable Tightness.- 5.9. Mappings of Compacta onto Tikhonov Cubes I?.- 5.10. Dyadic Compacta.- 5.11. Supercompacta and Extensions of the Class of Dyadic Compacta.- §6. Compact Extensions.- 6.1. General Remarks about Compact Extensions.- 6.2. Compact T1-Extensions.- 6.3. Embedding Topological Spaces into Compact T1-Spaces of Countable Weight.- 6.4. Compact Hausdorff Extensions, Relation of Subordination.- 6.5. Compact Extensions of Locally Compact Hausdorff Spaces.- 6.6. Duality Between Properties of a Space and of Its Remainder.- 6.7. Compact Extensions and Cardinal Invariants.- 6.8. Compact Hausdorff Extensions and Perfect Mappings.- 6.9. Properties of the ?ech-Stone Extension.- 6.10. Closing Remarks Concerning Compact Hausdorff Extensions.- §7. Compactness and Spaces of Functions.- 7.1. Natural Topologies on Spaces of Functions.- 7.2. Joint Continuity and Compact-Open Topology.- 7.3. Stone-Weierstrass Theorem.- 7.4. Convex Compact Sets and Krein-Milman Theorem.- 7.5. Theorem ofAlaoglu and Convex Hulls of Compacta.- 7.6. Fixed-Point Theorems for Continuous Mappings of Convex Compacta.- 7.7. Milyutin Compact Spaces.- 7.8. Dugundji Compact Spaces.- §8. Algebraic Structures and Compactness - A Review of the Most Important Results.- 8.1. Compacta and Ideals in Rings of Functions.- 8.2. Spectrum of a Ring. Zariski Topology.- 8.3. The Space of Maximal Ideals of a Commutative Banach Algebra.- 8.4. The Stone Space of a Boolean Algebra.- 8.5. Pontryagin's Duality Theory.- 8.6. Compact Extensions of Topological Groups. Almost Periodic Functions.- 8.7. Compacta and Namioka's Theorem About Joint Continuity of Separately Continuous Functions.- 8.8. Fragmentable and Strongly Fragmentable Compacta and Radon-Nikodým Compact Spaces.- 8.9. Hilbert Modules over C*-Algebras of Continuous Functions on Compacta.- 8.10. Compact Subsets of Topological Fields.- 8.11. Locally Compact Topological Groups and Paracompactness.- 8.12. Final Remarks.- References.