By Sam Nadler

A textbook for both a semester or yr path for graduate scholars of arithmetic who've had at the least one path in topology. Introduces continuum concept via a mix of classical and sleek innovations. Annotation copyright e-book information, Inc. Portland, Or.

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E) The set of limit points of the set of rational numbers is the entire real line. 1. (c) The following theorem explains the relationship between closed sets and limit points. 10: A subset A of P is dosed if and only if A contains all its limit points. Proof: 'Suppose first that A is dosed and consider a limit point x of A. If x were outside A, then P\A would be an open set containing x but containing no point of A, and we would be forced to condude that x is not a limit point of A. Thus, if x is a limit point of A, then it must be a member of A.

Then (a) A fl U B, = B U B,). 3. For a slightly different approach that may be easier to follow, a more detailed argument is given here for (a), showing that the sets A fl U,E, B, and (A fl B,) are subsets of each other. B,. Then x E A and x belongs to B, for some i E I. Suppose x E A fl (A fl B,). From this we conclude Thus x E A fl B,for some i, so x E Afl U B,CU (A flB,). ,EJ IEJ 18 ONE / INTRODUCIION (A fl Be). Then x E A fl B,for some For the reverse inclusion, suppose x E iEI. This meansthatxEAandxEB,forsomeiEl.

The generalization of open sets to P2 can easily be made once the subsets of P2 which are analogous to the open intervals of P are determined. An open interval in P centered at a is an interval (a — r, a + r), where r is a positive number, In terms of distance, (a — r, a + r) consists of all points x in P for which the distance from x to a is less than r: (a—r,a+r)={xEP:Ia—xI

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