By Vladimir V. Tkachuk

ISBN-10: 3319047469

ISBN-13: 9783319047461

ISBN-10: 3319047477

ISBN-13: 9783319047478

This paintings is a continuation of the 1st quantity released by way of Springer in 2011, entitled "A Cp-Theory challenge ebook: Topological and serve as Spaces." the 1st quantity supplied an creation from scratch to Cp-theory and normal topology, getting ready the reader for a qualified realizing of Cp-theory within the final portion of its major textual content. This current quantity covers a large choice of themes in Cp-theory and normal topology on the specialist point bringing the reader to the frontiers of contemporary learn. the quantity includes 500 difficulties and routines with whole suggestions. it might probably even be used as an advent to complex set thought and descriptive set conception. The e-book provides assorted issues of the idea of functionality areas with the topology of pointwise convergence, or Cp-theory which exists on the intersection of topological algebra, sensible research and normal topology. Cp-theory has a huge position within the category and unification of heterogeneous effects from those components of analysis. in addition, this booklet provides a pretty whole insurance of Cp-theory via 500 conscientiously chosen difficulties and routines. via systematically introducing all the significant themes of Cp-theory the e-book is meant to convey a committed reader from uncomplicated topological ideas to the frontiers of contemporary research.

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**Additional info for A Cp-Theory Problem Book: Special Features of Function Spaces**

**Example text**

232. Prove that any Cech-complete space is a p-space. Give an example of a ˇ p-space which is not Cech-complete. 233. Prove that the following conditions are equivalent for any space X : (i) for an arbitrary compactification bX of the space X , there exists a countable family of compact subspaces of bX which separates X from bX nX ; (ii) there exists a compactification bX of the space X and a countable family of compact subspaces of bX which separates X from bX nX ; (iii) there exists a compactification bX of the space X and a countable family of Lindelöf ˙-subspaces of bX which separates X from bX nX ; (iv) there exists a space Z such that X is a subspace of Z and there is a countable family of compact subspaces of Z which separates X from ZnX ; (v) X is a Lindelöf ˙-space.

Is called Alexandroff one-point compactification of the discrete space of cardinality Ä. Ä/ D Ä [ fag, where a … Ä. If x 2 Ä, let Bx D ffxgg. ÄnB/ W B is a countable subset of Äg. Ä/. Ä/ is called the Lindelöfication of the discrete space of cardinality Ä. g A such that fn ! f g. The closure operator u A ! X /. X /. X /. X /. X /. x/ for any x 2 X . As usual, the expression X ' Y says that the spaces X and Y are homeomorphic. A space X is metacompact if every open cover of X has a point-finite open refinement.

Suppose that X X D Y [Z, where Y and Z have countable tightness. X / Ä !. 411. Suppose that X X D Y [ Z, where Y and Z have countable weight. X / Ä !. 412. 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.

### A Cp-Theory Problem Book: Special Features of Function Spaces by Vladimir V. Tkachuk

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