The Second Midterm on Wednesday, November 18. Tips for the midterm:
Review all the definitions, statements and proofs of theorems, and examples in Chapters 2-4 of Rudin.
Review the basic concepts relating to topological spaces: the topology, the neighborhood filters, closed subsets, the closure operation, Hausdorff spaces.
Review basic concepts involving filters, filter bases, connection between sequences (nets) and the associated filters, how the concept of convergence of a sequence (net) translates into the corresponding concept of convergence for the associated filter.
Review various characterizations and consequences of compactness of a topological space; in particular, review the proof of the theorem stating that closed bounded subsets in the Euclidean space Rn are compact and vice-versa.
Review various characterizations and consequences of completeness of a metric space; review the proof that the Euclidean space Rn is a complete metric space.
Given a real-valued function f
defined on a subset of R, and a number ε>0, be able to
find a number δ>0, such that |x-y| < δ implies
|f(x)-f(y)| < ε (you should be able to do this for
polynomial, trigonometric, and the n-th-root functions).
The concepts related to uniform structures will not be tested.
Announcement: your most recent homework (the one you submitted on Monday) is already available from the envelope outside my office.
due
November 16 (Monday)
Do
all the exercises from Notes on
Uniform Structures.
due
November 6
Chapter 4:
1 2 3 4 5 (use the definition of continuity as in Rudin!)
And
master the Wikipedia article on Uniform
spaces.
due
October 30
Chapter 3:
16 20 21 22 23 24 25
Prove that a topological space X
is regular
if and only if, for any point x
є X, closed
neighborhoods form a basis of the filter of
all neighborhoods of x.
due
October 23
Chapter 3:
1 2 3 4 5 19
Chapter 2: 27 28 29 30
Prove that a compact
Hausdorff space is normal
(Hint: Prove first that it is regular.)
due
October 16
Chapter 2:
19 20 21 22 23 24 25 26
due
October 9
Chapter 2:
12 13 14 15 16 17 18
due
October 2
Chapter 2: 2
3 5 6 7 8 9 11
due
September 25
Do
exercises: 2 4 6 7 8 9 10 12 13 14 15 16 17 20 34 35 36 from the new
version of the Notes on Ordered Sets.
due
September 18
Do all
the exercises from Notes on Ordered
Sets.
due
September 11
Chapter
1: 6 7 8 9 10
due
September 4
Chapter 1:
1 2 3 4 5