By Vladimir V. Tkachuk

This paintings is a continuation of the 1st quantity released via Springer in 2011, entitled "A Cp-Theory challenge publication: Topological and serve as Spaces." the 1st quantity supplied an creation from scratch to Cp-theory and common topology, getting ready the reader for a certified figuring out of Cp-theory within the final component of its major textual content. This current quantity covers a large choice of issues in Cp-theory and normal topology on the specialist point bringing the reader to the frontiers of recent learn. the quantity comprises 500 difficulties and workouts with entire suggestions. it could even be used as an advent to complex set concept and descriptive set conception. The ebook provides diversified subject matters of the speculation of functionality areas with the topology of pointwise convergence, or Cp-theory which exists on the intersection of topological algebra, useful research and normal topology. Cp-theory has a major function within the class and unification of heterogeneous effects from those parts of study. in addition, this publication offers a fairly whole assurance of Cp-theory via 500 rigorously chosen difficulties and routines. by means of systematically introducing all of the significant subject matters of Cp-theory the booklet is meant to convey a committed reader from uncomplicated topological ideas to the frontiers of recent research.

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445. X /. 446. X /. 447. X /. 448.

Suppose that X is a separable space such that X X D Y [ Z, where Y and Z are metrizable. Prove that X is metrizable. 413. Suppose that X is a compact space such that X X D Y [ Z, where Y and Z are metrizable. Prove that X is metrizable. 414. Give an example of a non-metrizable space X such that X X is the union of two metrizable subspaces. S 415. Suppose that X ! g. , there is Y Xn such that there exists an open continuous map of Y onto X ! , and hence, there exists an open continuous map of Y onto X .

263. X / has the Baire property. Prove that X is countable. X / is Baire, then X is countable. 264. X / has the Baire property. 265. X / is a Lindelöf ˙-space and has the Baire property. Prove that X is countable. 266. X / is monolithic. 267. -stable. 268. Prove that any product and any -product of Lindelöf ˙-spaces is stable. -stable. 269. (Baturov’s theorem). Let X be a Lindelöf ˙-space. Y /. 270. Prove that every subspace of X is a Lindelöf ˙-space if and only if X has a countable network. 271.