By Vladimir V. Tkachuk

ISBN-10: 1441974415

ISBN-13: 9781441974419

ISBN-10: 1441974423

ISBN-13: 9781441974426

The conception 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 normal topology. Cp-theory has an incredible function within the class and unification of heterogeneous effects from each one of those parts of study. via over 500 rigorously chosen difficulties and workouts, this quantity offers a self-contained advent to Cp-theory and normal topology. by way of systematically introducing all of the significant themes in Cp-theory, this quantity is designed to convey a devoted reader from easy topological rules to the frontiers of contemporary study. Key positive factors comprise: - a distinct problem-based creation to the speculation of functionality areas. - unique options to every of the provided difficulties and workouts. - A complete 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 normal topology in addition to a reference consultant for experts learning Cp-theory and comparable themes. This booklet additionally presents various subject matters 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**

Prove that the Sorgenfrey line is not Cˇech-complete. Recall that the Sorgenfrey line is the space (R, ts), where ts is the topology generated by the family f[a, b) : a, b 2 R, a < bg as a base. 273. Prove that a second countable space is Cˇech-complete if and only if it embeds into Ro as a closed subspace. 274. Prove that ˇ ech-complete space has the Baire property. (i) Any C (ii) Any pseudocompact space has the Baire property. 275. Let X be a Baire space. Prove that any extension of X as well as any open subspace of X is a Baire space.

Prove that any extension of X as well as any open subspace of X is a Baire space. Show that a closed subspace of a Baire space is not necessarily a Baire space. 276. Prove that a dense Gd-subspace of a Baire space is a Baire space. As a consequence, Q is not a Gd-subset of R. 277. Prove that an open image of a Baire space is a Baire space. 278. Prove that Cp(X) is a Baire space if and only it is of second category in itself. Give an example of a non-Baire space Y which is of second category in itself.

157. Prove that, for any space X, if Y is a continuous image of X, then (i) (ii) (iii) (iv) (v) (vi) c(Y) d(Y) nw(Y) s(Y) ext(Y) l(Y) c(X). d(X). nw(X). s(X). ext(X). l(X). 158. Let ’ 2 fweight, character, pseudocharacter, i-weight, tightnessg. Show that there exist spaces X and Y such that Y is a continuous image of X and ’(Y) > ’(X). 159. Suppose that X is a space and Y & X. Prove that (i) (ii) (iii) (iv) (v) (vi) (vii) w(Y) nw(Y) c(Y) s(Y) iw(Y) t(Y) w(Y) w(X). nw(X). c(X). s(X). iw(X). t(X).

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

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