Assignment 2 - Solutions - MAT 327 - Summer 2014
Comprehension
[C.1] Let A X, a topological space. Prove that
A=
cfw_ C X : C is closed and A C
Solution for C.1. We show both containments:
[] Let p A. Then A is a closed set containing A that does not con
12 - Metric Spaces and Metrizable Spaces
1
Motivation
Metric Spaces are beautiful spaces. In the world of topology, metric spaces are the supermodels and R is the Cindy Crawford. When we ask the question How nice is this
topological space? we often mean H
Tutorial Problems - Sections 11 to 12 - MAT 327 - Summer
2014
8
Axiom of Choice
1. Fill in the details that Zorns Lemma implies every vector space has a basis.
2. Which direction of In a rst countable space, f is continuous is equivalent to xn x
implies f
Tutorial Problems - Sections 8 to 10 - MAT 327 - Summer
2014
8
Finite Products
1. Let T := cfw_U P(R) : 0 U cfw_, and let U := cfw_U P(R) : 1 U cfw_. Describe
the product topology on R R determined by T and U.
2. Let (X, T ) and (Y, U) be topological sp
Tutorial Problems - Sections 6 to 7 - MAT 327 - Summer
2014
6
Continuous Functions and Homeomorphisms
1. Give examples of topological spaces (X, T ) and (Y, U) and a function f : X Y
such that
(a) f is open but not closed.
(b) f is closed but not open.
2.
Tutorial Problems - Sections 1 to 3 - MAT 327 - Summer
2013
1
Topological Spaces
1. Let X be an innite set with p X and let Tp := cfw_ U P(X) : U = or p U .
Prove that Tp is a topology on X. (Called the particular point topology on X, at p.)
2. Is there a
Tutorial Problems - Sections 4 to 5 - MAT 327 - Summer
2014
4
Countability
1. Give an explicit bijection between N and the collection of all nite binary strings.
2. Prove, using countability, that there is transcendental number (i.e. a number that is
not
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
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 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
9 - Separation Axioms
1
Motivation
In this course we have seen the Hausdor property, and some places where it is useful (like
Assignment 4, A.1). Generally the Hausdor property will be an extra assumption we want
our topological spaces to have, but someti
8 - Finite Products
1
Motivation
In our ongoing eorts to make new topological spaces, we (briey) revisit the product
topology. We have already seen nite topological products, and we have seen how they
interact with various topological invariants (we know
Assignment 1 - Solutions - MAT 327 - Summer 2014
Comprehension
[C.1] Let ( X, T ) be a topological space, and let f : X Y be an injection.
Is U := cfw_ f [A] : A T a topology on the range of f ?
Answer for C.1. This is just straightforward application of
Assignment 4 - Solutions - MAT 327 - Summer 2014
Comprehension
1
[C.1] Let A := (, 0) cfw_ n : n N cfw_2 (3, 4] R, and give A the induced topology
from Rusual .
1. What is the interior (calculated in A) of cfw_2 (3, 4]?
1
2. Is cfw_ 2n : n N a convergen
Assignment 3 - Solutions - MAT 327 - Summer 2014
Comprehension
[C.1] Prove FIN(N) := cfw_ F N : F is nite is a countable set.
Proof 1 of C.1. Note that
P(cfw_ 0, 1, 2, . . . , n )
FIN(N) =
nN
and each of those power sets is nite, so we have written FIN(N
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
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
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
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
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
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
10 - Partial Orders, Linear Orders and Well-Orders
1
Motivation
Partial ordering is a very natural relation in mathematics, and in the real world. We often
want to sort out a bunch of data and order it somehow. A partial ordering is a basic
relation that
11 - Axiom of Choice
1
Motivation
What constitutes a rigourous mathematical argument? Normally we think of a proof as
being airtight if it starts from a true premise, then uses logical deductions to arrive at a new
statement. For example, in our proof tha
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