Math 544, Autumn 2007 - Notes

This page will include all handouts, selected homework problem solutions, and possibly some notes. Links are to files in pdf format.

• First day handout (as pdf). All the information from this handout is available elsewhere on this website.
• Solution to Exercise A.2(b). Let R be a relation on X, and let ~ be the equivalence relation generated by R. That is, let ~ be the intersection of all equivalence relations on X that contain R, Let x R' y mean "x R y or y R x." Show that x ~ y if and only if one or more of the following holds:
  (i) x = y; (ii) x R' y; or (iii) there is a finite sequence of elements z₁, z₂, ..., z_n, such that x R' z₁ R' z₂ ... R' z_n.

  Proof. Define a relation x S y if and only if one of the three conditions (i), (ii), or (iii) holds. We show below that S is an equivalence relation. Note that every equivalence relation that contains R must contain S, because the equivalence relation is reflexive; contains R and is symmetric; and is transitive. Then S contains R (by (ii)) and must be contained in every equivalence relation that contains R, so by definition S is equal to ~, as desired.

  Now we confirm that x S y is an equivalence relation. It is reflexive by (i). It is symmetric because both (ii) and (iii) are defined using R', which is symmetric by definition. It is transitive by (iii), because two such finite chains can be concatenated to make a new, longer but finite, chain. Thus S is an equivalence relation, completing the proof.

• Notes on paracompactness (as pdf), from lecture on Friday, 10/19, on the equivalence of second countability, paracompactnes, etc.
• Examples (as pdf) of quotient maps with various odd properties, and showing different combinations of connectedness properties.
• Solutions for Problem 4-5 and part of Problem 4-11. For each of these, I have included two proofs of one part of the problem, where the second proof comes from one or two of your colleagues.
• Solutions for Problem 5-5 and the "three circles" problem on the website for Week 6 homework.
• Solutions for Problem 8-3(b) and the hint in Problem 8-4, and outline of two different solutions for Problem 8-8.

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