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Math 131: Introduction to Topology 1
Professor Denis Auroux
Fall, 2019
Contents
9/4/2019 - Introduction, Metric Spaces, Basic Notions 3
9/9/2019 - Topological Spaces, Bases 9
9/11/2019 - Subspaces, Products, Continuity 15
9/16/2019 - Continuity, Homeomorphisms, Limit Points 21
9/18/2019 - Sequences, Limits, Products 26
9/23/2019 - More Product Topologies, Connectedness 32
9/25/2019 - Connectedness, Path Connectedness 37
9/30/2019 - Compactness 42
10/2/2019 - Compactness, Uncountability, Metric Spaces 45
10/7/2019 - Compactness, Limit Points, Sequences 49
10/9/2019 - Compactifications and Local Compactness 53
10/16/2019 - Countability, Separability, and Normal Spaces 57
10/21/2019 - Urysohn’s Lemma and the Metrization Theorem 61
1 Please email Beckham Myers at bmyers@college.harvard.edu with any corrections, questions, or comments. Any
mistakes or errors are mine.
10/23/2019 - Category Theory, Paths, Homotopy 64
10/28/2019 - The Fundamental Group(oid) 70
10/30/2019 - Covering Spaces, Path Lifting 75
11/4/2019 - Fundamental Group of the Circle, Quotients and Gluing 80
11/6/2019 - The Brouwer Fixed Point Theorem 85
11/11/2019 - Antipodes and the Borsuk-Ulam Theorem 88
11/13/2019 - Deformation Retracts and Homotopy Equivalence 91
11/18/2019 - Computing the Fundamental Group 95
11/20/2019 - Equivalence of Covering Spaces and the Universal Cover 99
11/25/2019 - Universal Covering Spaces, Free Groups 104
12/2/2019 - Seifert-Van Kampen Theorem, Final Examples 109
2
9/4/2019 - Introduction, Metric Spaces, Basic Notions
The instructor for this course is Professor Denis Auroux. His email is auroux@math.harvard.edu
and his office is SC539. He will be hosting office hours Monday 12:30-2 and Tuesday 9-10:30. The
course website is http://math.harvard.edu/ auroux/131f19/. All information will be posted on the
course webpage, although we will use Canvas to record grades.
There will be homework due every week on Wednesday, along with a take-home midterm and
an in class final. We will loosely follow Munkres’ Topology. The only prerequisites are some famil-
iarity with the notion of a group and some comfort with metric spaces/the ability to manipulate
open and closed sets.
Introduction
Broadly, geometry is the study of measuring quantities. Mathematicians then use these measure-
ments to make conclusions about properties of the spaces being studied. Topology, on the other
hand, studies spaces by asking questions from a qualitative perspective. For example, some topo-
logical questions include:
• Is a space connected?
• Is a space simply connected? This question provides a technique for distinguishing between
a sphere and a torus. For on the torus, there exist closed curves which cannot be ‘shrunk’ to
a point.
• Is a space oriented? For example, the regular cylinder is oriented (as it has two sides), while
the M¨obius space is not (it has only one side). Note that there are easier ways to distinguish
these two, namely by examining their boundaries.
Algebraic topology is the field that studies invariants of topological spaces that measure these above
properties. For example, the fundamental group measures how far a space is from being simply
connected. Before this, however, we will develop the language of point set topology, which extends
the theory to a much more abstract setting than simply metric spaces.
Today we will remain informal, but a topological space is an abstraction of metric spaces. In
short, a topological space is a set equipped with the additional data necessary to make sense of
what it means for points to be ‘close’ to each other. This will allow us to develop notions of limits
and continuity.
3
The Power of Abstraction - Example from Analysis
Wehave the following classical theorem:
Theorem (The Extreme Value Theorem). Given a continuous function f : [a,b] → R, f
achieves is maximum and minimum in the interval [a,b].
This theorem can be generalized to the following:
Theorem. Given a continuous function f : C → R from a compact set C, f achieves its
maximum and minimum in C.
And this is itself a special case of an even more general theorem:
Theorem. Given a continuous function f : C → X from a compact set C to a topological
space X, the image of f is compact.
This is one excellent example of the power of abstraction, as we can take existing results and
expand them to vastly more generalize situations.
Wewill introduce metric spaces in order to motivate the definition of topological spaces (otherwise,
the definition seems a bit arbitrary).
Metric spaces and open sets
Definition. A metric space is a pair (X,d), where X is a set and d : X × X → R≥0 is the
distance function. d should satisfy
1. d(x,x) = 0 and d(x,y) > 0 when x 6= y, for all x,y ∈ X.
2. d(x,y) = d(y,x), namely d is symmetric.
3. d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X. This is the triangle inequality, which says
that the shortest path between two points is the ‘straight line’ between them.
Examples
• The vector space Rn with the Euclidean distance
v
un
X
u 2
d(x,y) = t (y −x )
i i
i=0
where x = (x ,...,x ),y = (y ,...,y ) ∈ Rn, is a metric space. This is the usual
1 n 1 n
distance in space. It’s easy to check that this indeed defines a metric on the space Rn.
• Let Y ⊂ Rn. Then Y becomes a metric space under the induced metric. In particular,
we define a metric on Y by simply restricting the metric d|Y×Y on X.
4
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