This question came up recently, and I'm wondering whether or not it's true:
Let (X,A,m) be a finite measure space. Let E_1,E_2,... be a sequence of measurable subsets of X of constant positive measure (i.e., there exists c>0 such that m(E_i) = c for all i). Then there exists a subsequence of...
Can anyone provide any ideas or hints for this problem?
Let f:R^2 -> R satisfy the following properties:
- For each fixed x, the function y -> f(x,y) is continuous.
- For each fixed y, the function x -> f(x,y) is continuous.
- If K is a compact subset of R^2, then f(K) is compact...
I'm referring to question #26 in chapter 3 of Pugh's Real Mathematical Analysis.
For those without the book, here's the question:
Let X be a set with a transitive relation # (Note: #is just an abstract relation). It satisfies the condition that for all x1,x2,x3 in X, we have
x1 # x1
and
if x1...
Okay well, to start I figured I could just play around with the R^2 case.
For an open disk of radius r in R^2 (translated so its center is at the origin) I found that a both-ways continuous bijection from the disk to R^2 is
f(x,y) = (x,y) / (r-|(x,y)|)
Now to generalize a bit, I could try...
I'm having difficulty in proving that two spaces are homeomorphic; I understand the definition and such, but working out the details is not coming easy. For instance, our teacher asked us to prove that all convex open non-empty subsets of R^n are homeomorphic to R^n. How does one go about...
Homework Statement
Let U be a non-empty, convex, open subset of R^2. Prove that U is homeomorphic to R^2.
Hint: First prove that the intersection of a line in R^2 with U (if non-empty) is homeomorphic to an open interval in R^1. Then use radial projections.Homework Equations
We just have the...
I know Berkeley has a slight edge in the "prestige" factor, but is this edge well-earned? Is there a significant difference in the quality of instruction and opportunities for research at these two institutions, particularly in the pure math department? Is there much difference in the...
I took my sweet time, hitting the time button whenever necessary. Score came out to 167. Took it again without the time extensions, and the score was 198. LOL, that's 31 time extensions! But, slow and steady wins the race...right?