Spring 2014 Math 453
Homework Two
It is intended that you do not simply look up these proofs online. They are
there, but it wont help you learn to write a proof if you just look them up.
A closure operator on X is a function that maps the power set of X t
Spring 2014 Math 453
Homework Five Solutions
I am using Rolfsens table of knots found at the Knot Atlas:
http:/katlas.math.toronto.edu/
1. Determine the genus of the Figure Eight Knot.
We have 2 disks and 3 bands so we get a genus of
12+3
2
=1
2. Determin
Spring 2014 Math 453
Final
1. What types of things do topologists study? This should be a list, for
example, one thing that topologists study are knots. What other things
would go on this list?
Metric Spaces
Surfaces
Knots
Vector Fields
2. What is the
Winter 2014 Math 453
Homework One
For each line in each proof, mark whether:
N the statement has no meaning, the words are not being used correctly
F the statement is a false statement that is completely irredeemable
I the statement is a true statement, b
Spring 2014 Math 453
Homework One
For each line in each proof, mark whether:
N the statement has no meaning, the words are not being used correctly
F the statement is a false statement that is completely irredeemable
I the statement is a true statement, b
Spring 2014 Math 453
Homework Three Solutions
It is intended that you do not simply look up these proofs online. They are
there, but it wont help you learn to write a proof if you just look them up.
I am using Rolfsens table of knots found at the Knot Atl
Spring 2014 Math 453
Homework Two Solutions
It is intended that you do not simply look up these proofs online. They are
there, but it wont help you learn to write a proof if you just look them up.
A closure operator on X is a function that maps the power
Spring 2014 Math 453
Midterm
Since a lot of you are trying to use the notion of a topology to do problems with closure operators, were going to rename things to keep everything
straight. A klosure operator on X is a function that maps the power set of
X t
Spring 2014 Math 453
Midterm Solutions
1. Let X and Y be sets with klosure operators, prove that if f : X Y
is kontinuous, then for all A X, we have f (A) f (A).
Proof. Let A X, then f (A) f (A). Applying f 1 and noticing that
A f 1 (f (A), we have
A f 1
Spring 2014 Math 453
Homework Four Solutions
It is intended that you do not simply look up these proofs online. They are
there, but it wont help you learn to write a proof if you just look them up.
Recall that if A is a subset of a topological space, that