MAT327 INTRO TO TOPOLOGY
Books:
Topology, second edition
Counterexamples
in Topology
by J. Munkres
by L. Steen, J. Seebach
Instructor
Regina Rotman
Oce Hours
6262 Bahen Building
Friday 9-10, or by appointment
Grading Policy:
5 Homework Assignments
Midterm
Nets and filters (are better than sequences)
Contents
1 Motivation
2
2 More implications we wish would reverse
2
3 Nets
4
4 Subnets
6
5 Filters
9
6 The connection between nets and filters
12
7 The payoff
14
8 Filling in a gap from first year calculus
15
1
Countability
1
Motivation
In topology as well as other areas of mathematics, we deal with a lot of infinite sets. However,
as we will gradually discover, some infinite sets are bigger than others. Countably infinite
sets, while infinite, are small in a ve
Orders and 1
Contents
1 Motivation
2
2 Partial orders
2
3 Some terminology
4
4 Linear orders
5
5 Well-orders
9
6 1
11
7 1 + 1
13
1
1
Motivation
The main reason we are introducing this topic in this course is so that we can define order
topologies, and in
Subspaces
1
Motivation
In this section we will build perhaps our most useful tool for defining new topological spaces,
and for analyzing properties of the ones we define: how to define a topology on a subset of a
topological space in a way that agrees wit
Stronger separation axioms
1
Motivation
While studying sequence convergence, we isolated three properties of topological spaces that are
called separation axioms or T -axioms. These were called T0 (or Kolmogorov), T1 (or Frechet),
and T2 (or Hausdorff). T
Sequence convergence, the weak T-axioms,
and first countability
1
Motivation
Up to now we have been mentioning the notion of sequence convergence without actually defining
it. So in this section we will define sequence convergence in a general topological
Continuous functions and homeomorphisms
1
Motivation
Up to now we have defined just a few topological properties, like the first three T -axioms and the
countability properties (separable, ccc, first and second countable). More than anything else,
in my o
Finite Products
1
Motivation
This is the last part in our series exploring how to get new topological spaces from old ones. For
now, at least. We mentioned the definition of the product topology for a finite product way back
in Example 2.3.6 in the lectur
Bases of topologies
1
Motivation
In the previous section we saw some examples of topologies. We described each of them by
explicitly listing all of the open sets in each one. Unfortunately, this is not be a feasible strategy
for allor even mosttopologies
Connectedness
1
Motivation
Connectedness is the sort of topological property that students love. Its definition is intuitive
and easy to understand, and it is a powerful tool in proofs of well-known results.
Roughly speaking, a connected topological space
Compactness
1
Motivation
While metrizability is an analysts favourite topological property, compactness is surely a topologists favourite topological property. Metric spaces have many nice properties, like being first
countable, very separative, and so on
Tychonoff s theorem, and properties related
to compactness.
1
Motivation
In this section we will prove Tychonoffs theorem. Before we can do that, we will discuss a
different characterization of compactness in terms of filters and ultrafilters. We will hav
MAT 327 Homework assignment 1.
Due Monday, October 5, 2015
Each problem is worth 10 point
Problem 1. Let X be a topological space, and A be its subset. Is it true that
A , the set of all limit points of A is always closed in X?
Problem 2. Show that if A i
MAT 327 Homework assignment 2.
Due Thursday, October 15, 2015
Each problem is worth 10 points
Problem 1. Let x1 , x2 , . be a sequence of points of X with the box topology.
Is it true that this sequence converges to the point x if and only if the sequence
MAT 327 Sample Midterm Exam.
October, 2015
The total number of points for this exam is 350.
Problem 1 [40 points].
Which of the following spaces are Hausdor?
(a) lR with the discrete topology.
(b) lR with the trivial topology.
(c) lR with the product topo
The Axiom of Choice
Contents
1 Motivation
2
2 The Axiom of Choice
2
3 Two powerful equivalents of AC
4
4 Zorns Lemma
5
5 Using Zorns Lemma
6
6 More equivalences of AC
11
7 Consequences of the Axiom of Choice
12
8 A look at the world without choice
14
1
1
Arbitrary products
Contents
1 Motivation
2
2 Background and preliminary definitions
2
3 The space RN
box
4
4 The space RN
prod
6
5 The uniform topology on RN , and completeness
12
6 Summary of results about RN
14
7 Arbitrary products
14
8 A final note on
Metric spaces and metrizability
1
Motivation
By this point in the course, I hope this section should not need much in the way of motivation.
From the very beginning, we have talked about Rnusual and how relatively easy it is to prove
things about it due t
Things You Should Know
This is a slightly modified version of a document authored by Micheal Pawliuk. This list of concepts is
used here with his permission.
1
Basic Set Theory
I will assume students are familiar with all of these terms and symbols. Pleas
Urysohns metrization theorem
1
Motivation
By this point in the course, I hope that once you see the statement of Urysohns metrization
theorem you dont feel that it needs much motivating. Having studied metric spaces in detail
and having convinced ourselve
Urysohns Lemma
1
Motivation
Urysohns Lemma (it should really be called Urysohns Theorem) is an important tool in topology. It will be a crucial tool for proving Urysohns metrization theorem later in the course,
which is a theorem that provides conditions
Closed sets, closures, and density
1
Motivation
So far in this course, all we have done is define what topologies are, define a way of comparing
two topologies, define a method for more easily specifying a topology (as a collection of sets
generated by a
Introduction
1
Motivation and foreshadowing
Most of the fundamental concepts in this course are generalizations of concepts with which you
are familiar from previous courses in analysis and/or linear algebra. For example, you have
likely encountered of at
4 - Countability Review
1
Topological Motivation
In many branches of mathematics, the notion of innity arises. As is turns out, not all
innities are created equal. For us countable sets will often be thought of as quite small
(and innite), whereas uncount
5 - Convergence and Limit points
1
Motivation
So far we have seen how to describe when a point is close to a set; we used the notion of
the closure of a set to describe closeness. This has the advantage that it makes sense in
every topological space, and
1 - Introduction to Topologies
1
Some Motivating Questions
Here are some things to chew on:
What does the mean in the sentence xn x?
What does it mean to say that two sets of points are close to each other? Is (, 0)
close to (0, +)? Is close to Q?
When
3 - Closed Sets and Closures
1
So Far .
So far we have seen a couple examples of topological spaces (R with the usual topology,
discrete topol- hey, why am I listing them? You should be the one trying to remember
the 8 or so topologies weve discussed so f
6 - Continuous Functions and Homeomorphisms
1
Motivation
There is an old joke (that isnt particularly funny) that goes: A topologist is a person
who cant tell the dierence between a doughnut and a coee cup. The idea here is that
a topologist thinks that t
2 - Basis
1
Motivation
We have already seen some topologies, but there are many more out there! The problem
is that it is often dicult (or impossible!) to list out all of the open sets in a topology.
Here we will use the notion of a basis, which will be a