MATH 131 SECTION NOTES, OCTOBER 27
1. Midterm solutions
1.1. Problem 1.
(a) FALSE. If that was true, 1-to-1 wouldnt be a very useful concept, would it? Consider any
constant function f from a set with more than one element to itself, then f clearly fails
MATH 131 SECTION, IV
0.1. Notation. Throughout we denote by X a metric space, and by dX the metric on X.
1. Preliminaries
Lets rst remember what convergent and Cauchy sequences and completeness are in
metric spaces.
1.1. Denition. A sequence (xk ) X N is
MATH 131 SECTION NOTES, OCTOBER 6
Today Id like to talk about something that I personally nd interesting - some of the (unexpected)
combinatorial applications of topology.
1. Furstenbergs proof of the infinitude of primes
We start with a now classical pro
MATH 131 SECTION NOTES, SEPTEMBER 15
1. Uniqueness of limits and the Hausdorff assumption
Remember that in class we saw that if we give any innite set X the conite topology, any
sequence of distinct elements in X converges to any point in X! Pathological
MATH 131 SECTION NOTES, SEPTEMBER 22
1. Connected components
Denition 1.1. Given a topological space X, we dene an equivalence relation by letting x y
i there is a connected subspace of X containing both x and y. Then the equivalence classes with
respect
MATH 131 SECTION NOTES, SEPTEMBER 29
1. Some notes on the homework
1.1. The balls problem via compactness. Theres a super easy way to do the balls problem
with compactness! Recall the problem:
Problem 1.1. Let U Rn be an open, connected set. Prove that fo
MATH 131 SECTION NOTES, OCTOBER 13
1. Group theory review
Denition 1.1. A group is a set G with a binary operation (also called group operation or
composition law) : G G G denoted multiplicatively by (a, b) a b, satisfying the following
properties:
(1) Th
MATH 131 SECTION NOTES, NOVEMBER 3
1. Cayley graphs
Denition 1.1. For a group G, a generating set S G is a subset such that every g G can
be expressed in terms of the group operation on nitely many elements of S S 1 .
Denition 1.2. A directed graph G = (V
MATH 131 SECTION NOTES, NOVEMBER 10
1. Graphs and topology
To dene a graph as a topological space formally, we rst need to dene the disjoint union
topology:
Denition 1.1. Let cfw_X be topological spaces, and let
X=
X
be their disjoint union, with : X X t
MATH 131 SECTION NOTES, OCTOBER 20
1. (Some) Midterm review
1.1. Set theory. What is a set (trick question)? What is a map between sets? Properties: injective, surjective, bijective. When is a set said to be nite? If A, B are sets, when do we say that
|A|
MATH 131 SECTION, II
1. Apologies
Professor McMullen has mentioned a few times that one of the reasons we talk about everything in terms of open and closed sets in general topology is that the notion of convergent
sequences doesnt capture all phenomena in