Spring 2010
Analysis (MATH 305b)
NOTE: No Office Hours on the following Fridays: Feb 19, 26, March 26, April 9 (I am attending the CUNY Seminar on Commutative Algebra and Algebraic Geometry on these days.)
Here is what we covered on:
Monday, January 11th: Review of Chapters 1-4
Wednesday, January 13th: Review of Chapters 5-8
Friday, January 15th: Discussion of Compactness, completeness, boundedness
Wednesday, January 20th: Sections 10A and 10B
Friday, January 22nd: Section 10C (first half)
Monday, January 25th: Sections 10C and 10D
Wednesday, January 27th: finished 10D and 10E
Friday, January 29th: Section 11A (first half)
Monday, February 1st: Section 11A
Wednesday, February 3rd: Section 11C and half of 11D
Friday, February 5th: Section 11D
Monday, February 8th: Section 11D and 11E
Wednesday, February 10th: 11E (Peter Jones)
Friday, February 12th: 12A
Monday, February 15th: 12B
Friday, February 19th: 12B
Monday, February 22nd: 12C
Wednesday, February 24th: Review
Friday, February 26th: Midterm Exam
Monday, March 1st: 12D
Wednesday, March 3rd: Discussion of the Midterm Exam
Friday, March 5th: 13A
Monday, March 22nd: 13B
Wednesday, March 24th: 13C
Friday, March 26th: 13D and E
Monday, March 29th: 13F
Wednesday, March 31st: 13G and H
Friday, April 2nd: 14A and B
Monday, April 5th: 14 C
Wednesday, April 7th: 14G and 15A
Friday, April 9th: 15B
Monday, April 12th: 15C
Wednesday, April 14th: 15D
Monday, April 19th: 15E and 15F
Homework
due Friday, January 29: Section 10B #3,5
Section 10C #1,2
Read Chapter 10
Bonus Problem 1: Let (X,d) be a metric space.
Show that a subset A of X is compact if and only if it is totally bounded and complete.
Bonus Problem 2: Find an open cover of the unit interval (0,1) without any finite subcover. (standard euclidean
metric on the reals)
due Friday, February 5: Section 10D # 1,4,5
Section 11A # 1,3,4
Read Chapter 11
Bonus Problem 1: Fix a prime p, for any two different integers n,m define d(n,m)= 2^{-k}
where k is the maximal integer such that p^k divides |n-m|.
Set d(n,n)=0. Show that d defines a metric on the integers. Show also
that every point is closed and not open w.r.t. this metric.
Bonus Problem 2: Show that the union of the B_n's in the proof of the approximation property
is closed.
due Friday, February 12: Section 11D # 2,3,4,10,11
Read Chapter 11
Bonus Problem 1: Consider the reals with the euclidean metric d. Define delta= d/(1+d)
Show that delta is also a metric on the reals. Show also that w.r.t delta
the reals are bounded but not totally bounded.
Bonus Problem 2: Let X be a countable set, X={x_i | i=1,2,3,...} define
d(x_i,x_j) = 1 + 1/(i+j) if i and j are different
0 otherwise
Show that d is a metric on X. Show that each point in X is open. Show also
that every Cauchy sequences becomes stationary.
due Friday, February 19: Section 11E # 3,4,8
Section 12B # 3,4
Read Chapter 12
Let X be a set, and T a subset of the power set of X. We say that T defines a topology on X if
T contains the empty set, the whole set X, is closed under arbitrary unions and finite intersections.
The sets in T are called the open sets of X.
Consider the complex numbers. Show that the following choices for T indeed define topologies:
Bonus Problem 1: (Zariski topology) U is open if it is the complement of the zero set of a polynomial.
Bonus Problem 2: (Fort topology) U is open if its complement is finite or it does not contain the origin.
due Monday, March 1: Section 12C # 1,2
Section 12D # 4,7
Read Chapter 12
Let (X, T) and (Y,S) be topological spaces. we call a map f: X --> Y continuous if f^{-1}(s) \in T
for all s\in S.
Bonus Problem 1: Classify the continuous maps on a set X equipped with the discrete topology.
Bonus Problem 2: Classify the continuous maps on a set X equipped with the indiscrete topology.
you may hand this homework set in after spring break!
due Monday, March 22: Page 172, # 1,2,5,6
Read Chapter 13
Bonus Problem 1: Let X be a set equipped with two different topologies T and S. Under which assumptions
is the identity map id: (X,T)---> (X,S) continuous?
Bonus Problem 2: Prove that in a Hausdorff space every convergent sequence has a unique limit point. How about
the converse?
Bonus Problem 3: Consider the space of bounded continuous functions f:[0,1] ---> R. Define a metric on that space
by d(f,g)= sup { |f(x)-g(x)| , x\in R}. Is the so obtained metric topology on this space equivalent to the
compact-open topology?
due Friday, March 26: Section 13B # 1,2,5,9
due Friday, April 2: Section 13C # 1,3
Section 13D #1,2
Section 13E #1
Due Friday, April 9: Section 13F #2,4
Section 13G #1,2,5
a topological space X is disconnected if there are proper open subsets U, V in X such that U\cap V is empty and U\cup V=X,
Otherwise the space is called connected.
Bonus Problem 1: Show that X is connected if and only if
X=A\cup B, A and B nonempty implies that A\cap closure(B) or closure(A)\cap B is not empty
Bonus Problem 2: Show that the image of a continuous map with a connected domain is connected.
due Friday, April 16: Section 14A #2
Section 14B # 1
Section 14G #3
Section 15A # 1
Section 15B #2,5
Bonus Problem 1: Show that a path connected topological space is connected.
Bonus Problem 2: Show that the topological sine curve is connected but not path connected:
{ (x, sin(1/x)) | 0<x} \cup {(0,0)}
(considered as a subspace in R^2 with the usual metric topology)
due Monday, April 26: Section 15C #5,8
Section 15D #2
Section 15E #1,4
Section 15F #1
Bonus Problem 1: Show that a topological space G together with a group structure is a topological group
if and only if for any two g,h \in G and any open neighborhood U(gh^{-1}) there exist
neighborhoods U(g) and U(h) such that U(g)U(h)^{-1} \subseteq U(gh^{-1})
Bonus Problem 2: If G is a topological group than the map F:G ---> G, g --> F(g)=g^{-1} is a homeomorphism,
i.e., F is bijective and continuous and F^{-1} is continuous too.
Exams
Midterm Exam, in class
Final Exam Monday, May 10 9am