Topological Function Spaces

Inhaltsverzeichnis

• 0. General information on Cp(X) as an object of topological algebra. Introductory material.
• 1. General questions about Cp(X).
• 2. Certain notions from general topology. Terminology and notation.
• 3. Simplest properties of the spaces Cp(X, Y).
• 4. Restriction map and duality map.
• 5. Canonical evaluation map of a space X in the space CpCp(X).
• 6. Nagata’s theorem and Okunev’s theorem.
• I. Topological properties of Cp(X) and simplest duality theo-rems.
• 1. Elementary duality theorems.
• 2. When is the space Cp(X) u-compact?.
• 3. “tech completeness and the Baire property in spaces Cp(X).
• 4. The Lindelöf number of a space Cp(X), and Asanov’s theorem.
• 5. Normality, collectionwise normality, paracompactness, and the extent of Cp(X).
• 6. The behavior of normality under the restriction map between function spaces.
• II. Duality between invariants of Lindelöf number and tightness type.
• 1. Lindelöf number and tightness: the Arkhangel’skii—Pytkeev theorem.
• 2. Hurewicz spaces and fan tightness.
• 3. Fréchet—Urysohn property, sequentiality, and the k-property of Cp(X).
• 4. Hewitt—Nachbin spaces and functional tightness.
• 5. Hereditary separability, spread, and hereditary Lindelöf number.
• 6. Monolithic and stable spaces in Cp-duality.
• 7. Strong monolithicity and simplicity.
• 8. Discreteness is a supertopological property.
• III. Topological properties of function spaces over arbitrary compacta.
• 1. Tightness type properties of spaces Cp(X), where X is a compactum, and embedding in such Cp(X).
• 2. Okunev’s theorem on the preservation of Q-compactness under t-equivalence.
• 3. Compact sets of functions in Cp(X). Their simplest topological properties.
• 4. Grothendieck’s theorem and its generalizations.
• 5. Namioka’s theorem, and Pták’s approach.
• 6. Baturov’s theorem on the Lindelöf number of function spaces over compacta.
• IV. Lindelöf number type properties for function spaces over compacta similar to Eberlein compacta, and properties of such compacta.
• 1. Separating families of functions, and functionally perfect spaces.
• 2. Separating families of functions on compacta and the Lindelöf number of Cp(X).
• 3. Characterization of Corson compacta by properties of the space Cp(X).
• 4. Resoluble compacta, and condensations of Cp(X) into a ?*-product of real lines. Two characterizations of Eberlein compacta.
• 5. The Preiss—Simon theorem.
• 6. Adequate families of sets: a method for constructing Corson compacta.
• 7. The Lindelöf number of the space Cp(X), and scattered compacta.
• 8. The Lindelöf number of Cp(X) and Martin’s axiom.
• 9. Lindelöf ?-spaces, and properties of the spaces Cp, n(X).
• 10. The Lindelöf number of a function space over a linearly ordered compactum.
• 11. The cardinality of Lindelöf subspaces of function spaces over compacta.