A Cp-Theory Problem Book: Compactness in Function Spaces by Vladimir V. Tkachuk

By Vladimir V. Tkachuk

Discusses a large choice of top-notch tools and result of Cp-theory and basic topology provided with designated proofs
Serves as either an exhaustive path in Cp-theory and a reference consultant for experts in topology, set concept and useful analysis
Includes a finished bibliography reflecting the cutting-edge in smooth Cp-theory
Classifies a hundred open difficulties in Cp-theory and their connections to past study

This 3rd quantity in Vladimir Tkachuk's sequence on Cp-theory difficulties applies all smooth equipment of Cp-theory to check compactness-like homes in functionality areas and introduces the reader to the idea of compact areas established in sensible research. The textual content is designed to deliver a committed reader from simple topological rules to the frontiers of recent study masking a wide selection of themes in Cp-theory and common topology on the specialist level.

The first quantity, Topological and serve as areas © 2011, supplied an advent from scratch to Cp-theory and basic topology, getting ready the reader for a qualified realizing of Cp-theory within the final component of its major textual content. the second one quantity, targeted positive aspects of functionality areas © 2014, endured from the 1st, giving kind of entire assurance of Cp-theory, systematically introducing all of the significant themes and delivering 500 conscientiously chosen difficulties and workouts with whole strategies. This 3rd quantity is self-contained and works in tandem with the opposite , containing rigorously chosen difficulties and ideas. it may well even be regarded as an creation to complicated set concept and descriptive set concept, proposing assorted issues of the idea of functionality areas with the topology of aspect clever convergence, or Cp-theory which exists on the intersection of topological algebra, sensible research and normal topology.

Algebraic Topology
Functional research

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Construct an Eberlein compact space which does not have a closure-preserving cover by compact metrizable subspaces. 368. Observe that every strong Eberlein compact is Eberlein compact. Prove that a metrizable compact space is strong Eberlein compact if and only if it is countable. 369. Prove that a compact X is strong Eberlein compact if and only if it has a point-finite T0 -separating cover by clopen sets. 370. Prove that every -discrete compact space is scattered. Give an example of a scattered compact non- -discrete space.

218. (Okunev’s theorem). X /. Y / is a Lindelöf ˙-space. 219. X / be Lindelöf ˙-spaces. X / is a Lindelöf ˙-space. X / is Lindelöf ˙ then all iterated function spaces of X are Lindelöf ˙-spaces. 220. X / is Gul’ko compact. 221. X / is a Lindelöf ˙-space. X / is Gul’ko compact. 222. , M is Talagrand compact; (ii) there is x 2 M such that M nfxg is pseudocompact and M is the Stone– ˇ Cech extension of M nfxg. 223. 224. 225. 226. 227. 228. 229. 230. X / is not countably compact. Suppose that, for a countably compact space X , there exists a condensation f W X !

317. Let X be a metrizable space. X / is functionally perfect if and only if X is second countable. 318. Prove that any paracompact space with a Gı -diagonal can be condensed onto a metrizable space. Deduce from this fact that any paracompact space with a Gı -diagonal is functionally perfect. 319. Observe that any Eberlein–Grothendieck space is functionally perfect. Give an example of a functionally perfect space which is not Eberlein–Grothendieck. 320. Prove that every metrizable space embeds into an Eberlein compact space.

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