By Vladimir V. Tkachuk

The conception of functionality areas endowed with the topology of pointwise convergence, or Cp-theory, exists on the intersection of 3 vital parts of arithmetic: topological algebra, sensible research, and basic topology. Cp-theory has an immense function within the class and unification of heterogeneous effects from every one of those components of study. via over 500 conscientiously chosen difficulties and workouts, this quantity offers a self-contained creation to Cp-theory and basic topology. via systematically introducing all the significant subject matters in Cp-theory, this quantity is designed to carry a devoted reader from uncomplicated topological ideas to the frontiers of contemporary study. Key good points contain: - a different problem-based creation to the speculation of functionality areas. - particular suggestions to every of the offered difficulties and routines. - A complete bibliography reflecting the cutting-edge in smooth Cp-theory. - various open difficulties and instructions for extra study. This quantity can be utilized as a textbook for classes in either Cp-theory and normal topology in addition to a reference consultant for experts learning Cp-theory and similar subject matters. This publication additionally offers various themes for PhD specialization in addition to a wide number of fabric compatible for graduate research.

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**Example text**

Iii) A finite set of maps F is uniformly equicontinuous if and only if each f 2 F is uniformly continuous. 247. Let (X, d) be a compact metric space. Given a metric space (Y, r) and an equicontinuous family F & C(X, Y), prove that F is uniformly equicontinuous. 248. Suppose that X is a space and (Y, r) is a (complete) metric space. For any functions f, g 2 C*(X, Y) let s(f, g) ¼ supfr(f(x), g(x)) : x 2 Xg. Show that s is a (complete) metric on C*(X, Y). It is called the metric of uniform convergence.

Prove that we have jXj ¼ w(Cp(X)) ¼ w(Cp(X)) for any infinite space X. In particular, weight of Cp(X) is countable if and only if X is countable. 170. Let X be an arbitrary space. Suppose that there exists a compact subspace K of Cp(X) such that w(K, Cp(X)) o. Prove that X is countable. 171. Given a space X and x 2 X, call a family B & t*(X) a p-base of X at x, if for any U 2 t(x, X) there is V 2 B such that V & U. Note that the elements of a p-base at x need not contain the point x. Prove that if Cp(X) has a countable p-base at some of its points then X is countable.

349. Suppose that there exists a strongly dense subspace A & Cp(X) with c(A) o. Is it true that Cp(X) has countable pseudocharacter? 350. Suppose that there exists a strongly dense s-pseudocompact subspace A & Cp(X). Prove that X is pseudocompact. 351. Suppose that there exists a strongly dense s-countably compact set A & Cp(X). Prove that X is compact. 352. Suppose that there exists a strongly dense countable A & Cp(X). Prove that X is compact and metrizable. 353. Give an example of a non-compact space X for which there exists a strongly dense s-pseudocompact A & Cp(X).