HW21
1. Prove that any set of triangles in R2 whose vertices have rational coordinates is countable.
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Solution. Let T be a triangle with vertices (r1 ; r1 ) ; (r2 ; r2 ) and
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(r3 ; r3 ), where rj ; rj 2 Q, j = 1; 2; 3. We can always assume that
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1. (a) Prove that, if a subset E in metric space (M; ) is of the rst category and A E , then A is also of the rst category.
(b) Prove that if (En )1
(M; ) is a sequence of sets of the rst
n=1
1
category, then E = [n=1 En is also of the rst category.
2. (a
Real analysis for graduate
students
Version 3.1
Richard F. Bass
January 6, 2016
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c Copyright 2016 Richard F. Bass
All rights reserved.
First edition published 2011.
Second edition published 2013.
Version 3.1 published 2016.
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To the memory of my paren
Real Analysis - Homework solutions
Chris Monico, May 2, 2013
1.1 (a) Rings (resp. -rings) are closed under finite (resp. countable) intersections.
(b) If R is a ring (resp. -ring) then R is an algebra (resp. -algebra) iff X R.
(c) If R is a (nonempty) -ri
HW2
1. Prove that any set of triangles in R2 whose vertices have rational coordinates is countable.
Note: Let x1 ; x2 ; x3 2 R2 . Then the triangle T with vertices x1 ; x2 ; x3
is the union of points of the segments [x1 ; x2 ] ; [x2 ; x3 ] and [x1 ; x3 ].
onto
1. (a) Let (A; ) be a well-ordered set, B
A and f : A ! B be an
order isomorphism of (A; ) onto (B; ). Prove that for every a 2 A,
a f (a).
(b) Let (A; ) be a well-ordered set and a 2 A. Recall that the set
Ia = fx 2 A j x
ag
is called an initial seg
HW 31
onto
1. (a) Let (A; ) be a well-ordered set, B
A and f : A ! B be an
order isomorphism of (A; ) onto (B; ). Prove that for every a 2 A,
a f (a).
(b) Let (A; ) be a well-ordered set and a 2 A. Recall that the set
Ia = fx 2 A j x
ag
is called an initi
HW4
1. Let X be an uncountable set and R X be the family of all sets which
either countable or their complements (in X) are countable. Prove that
R is a ring.
2. Describe the minimal ring generated by F in the following cases:
Describe the minimal ring ge