Compactness is related to a number of fundamental concepts of mathemat ics. Particularly important are compact Hausdorff spaces or compacta. Com pactness appeared in mathematics for the first time as one of the main topo logical properties of an interval, a square, a sphere and any closed, bounded subset of a finite dimensional Euclidean space. Once it was realized that pre cisely this property was responsible for a series of fundamental facts related to those sets such as boundedness and uniform continuity of continuous func tions defined on them, compactness was given an abstract definition in the language of general topology reaching far beyond the class of metric spaces. This immensely extended the realm of application of this concept (including in particular, function spaces of quite general nature). The fact, that general topology provided an adequate language for a description of the concept of compactness and secured a natural medium for its harmonious development is a major credit to this area of mathematics. The final formulation of a general definition of compactness and the creation of the foundations of the theory of compact topological spaces are due to P.S. Aleksandrov and Urysohn (see Aleksandrov and Urysohn (1971)).
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ISBN-13
978-3-642-77030-2 (9783642770302)
DOI
10.1007/978-3-642-77030-2
Schweitzer Klassifikation
§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.
