By Vladimir V. Tkachuk

ISBN-10: 1441974415

ISBN-13: 9781441974419

ISBN-10: 1441974423

ISBN-13: 9781441974426

The thought of functionality areas endowed with the topology of pointwise convergence, or Cp-theory, exists on the intersection of 3 very important parts of arithmetic: topological algebra, useful research, and basic topology. Cp-theory has a huge function within the type and unification of heterogeneous effects from each one of those parts of analysis. via over 500 conscientiously chosen difficulties and workouts, this quantity presents a self-contained creation to Cp-theory and common topology. by means of systematically introducing all of the significant themes in Cp-theory, this quantity is designed to convey a committed reader from simple topological rules to the frontiers of contemporary study. Key gains comprise: - a different problem-based advent to the idea of functionality areas. - precise recommendations to every of the offered difficulties and workouts. - A complete bibliography reflecting the state of the art in smooth Cp-theory. - a variety of open difficulties and instructions for additional study. This quantity can be utilized as a textbook for classes in either Cp-theory and common topology in addition to a reference advisor for experts learning Cp-theory and comparable issues. This publication additionally offers a number of issues for PhD specialization in addition to a wide number of fabric appropriate for graduate research.

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**Extra info for A Cp-Theory Problem Book: Topological and Function Spaces**

**Example text**

Y is a pseudo-open map then Y is Fre´chet–Urysohn. (v) A space is Fre´chet–Urysohn if and only if it is a pseudo-open image of a metrizable space. 226. Prove that a perfect image of a metrizable space is a metrizable space. 227. Show that a closed image of a countable second countable space is not necessarily a metrizable space. 228. , there is a metrizable space M and a closed map ’ : M ! Cp(X)). Prove that Cp(X) is metrizable and hence X is countable. 229. , there is a metrizable space M and an open map ’ : M !

If X is a space, and (Y, r) is a metric space, a map f : X ! Y is called bounded if f(X) is a bounded subset of Y. The set of all continuous bounded maps from X to Y is denoted by C*(X, Y). For an arbitrary A & C(X), denote by Au the set ff 2 C(X) : there exists a sequence ffn : n 2 og & A such that fn ! fg. Thereu exists a topology tu called the uniform convergence topology on C(X) such that A ¼ cltu ðAÞ for every A & C(X). The space (C(X), tu) will be denoted Cu(X). If C*(X) has the topology inherited from Cu(X), it is denoted CÃu ðXÞ.

343. Prove that there exist countable spaces X for which Cp(X) is not a Ksd-space. 344. Call a subset A & Cp(X) strongly (or uniformly) dense if, for every f 2 Cp(X), there is a sequence ffn : n 2 og & A such that fn ! f. In other words, a subset is strongly dense in Cp(X) if it is dense in the uniform convergence topology on C(X). Prove that (i) If A & Cp(X) is strongly dense in Cp(X) then it is dense in Cp(X). 4 Compactness Type Properties in Function Spaces 39 (ii) For any compact X, the space Cp(X) has a strongly dense s-compact subspace if and only if it has a dense s-compact subspace.

### A Cp-Theory Problem Book: Topological and Function Spaces by Vladimir V. Tkachuk

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