By Vladimir V. Tkachuk
The thought of functionality areas endowed with the topology of pointwise convergence, or Cp-theory, exists on the intersection of 3 vital components of arithmetic: topological algebra, sensible research, and basic topology. Cp-theory has an immense position within the class and unification of heterogeneous effects from every one of those parts of study. via over 500 conscientiously chosen difficulties and workouts, this quantity presents a self-contained advent to Cp-theory and normal topology. via systematically introducing all the significant themes in Cp-theory, this quantity is designed to convey a devoted reader from easy topological rules to the frontiers of recent study. Key beneficial properties contain: - a different problem-based creation to the idea of functionality areas. - specific strategies to every of the awarded difficulties and workouts. - A finished bibliography reflecting the cutting-edge in glossy Cp-theory. - quite a few open difficulties and instructions for extra examine. 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 similar themes. This e-book additionally presents quite a few issues for PhD specialization in addition to a wide number of fabric appropriate for graduate research.
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Additional info for A Cp-Theory Problem Book: Topological and Function Spaces
185. (The Dini theorem). Let X be a pseudocompact space. Suppose that fn 2 Cp(X) and fnþ1(x) ! fn(x) for all x 2 X and n 2 o. Prove that if there exists f 2 Cp(X) such that fn ! f then the sequence ffng converges uniformly to the function f. 186. Prove that the following are equivalent for any non-empty space X: Cp(X) is s-compact. Cp(X) is s-countably compact. , every f 2 Cp(X) has a compact neighbourhood. , every f 2 Cp(X) has a countably compact neighbourhood. , every f 2 Cp(X) has a pseudocompact neighbourhood.
320. Prove that Cp(o1 þ 1) is not normal. 321. Let X be an arbitrary space. Supposing that all compact subspaces of Cp(X) are first countable, prove that they are all metrizable. 322. Does there exist a space X such that all countably compact subspaces of Cp(X) are first countable but not all of them are metrizable? 323. Suppose that all countably compact subspaces of Cp(X) are metrizable. Is the same true for all pseudocompact subspaces of Cp(X)? 324. Is it true that any compact space X can be embedded into Cp(Y) for some pseudocompact space Y?
In particular, w(X) ¼ c(X). 328. Let X be a space. Call a set F ¼ fxa : a < kg & X a free sequence of length k if fxa : a < bg \ fxa : a ! bg ¼ ; for every b < k. Prove that, for any compact space X, tightness of X is equal to the supremum of the lengths of free sequences in X. 329. Prove that jXj 2w(X) for any compact space X. In particular, the cardinality of a first countable compact space does not exceed c. 330. Given an infinite cardinal k, let X be a compact space such that w(x, X) ! k for any x 2 X.