Recent content by GridironCPJ

Hello all, I'm having trouble showing that |e^it|=1, where i is the imaginary unit. I expanded this to |cos(t)+isin(t)| and then used the definition of the absolute value to square the inside and take the square root, but I keep getting stuck with √(cos(2t)+sin(2t)). Does anyone have any...

Yes, I realized earlier that my expansion was incorrect. I used polar coordinates, so we start with ∫∫[(rcosθ+irsinθ)^j(rcosθ-rsinθ)^k]rdrdθ From here I integrated with respect to r first and then I pulled in out and was left with [r^(j+k+2)/j+k+2]∫(cosθ+isinθ)^j(cosθ-isinθ)^k dθ...

Hello all, I'm working through a majority of the problems in "A First Course in Wavelets with Fourier Analysis" and have stumbled upon a problem I'm having difficulty with. Please view the PDF attachment, it shows the problem and what I have done with it so far. Once you have seen the...

Homework Statement Find an explicit homeomorphism from (0,1) to R. Homework Equations A homeomorphism from (-1,1) to R is f(x)=tan(pi*x/2). The Attempt at a Solution I'm horrible a modifying trig functions. Obviously, to shift by b you add b to (x) and you can change the...

Whoops, I forgot to mentione that NxNx[0,1) has the dictionary order topology. Does this change your mind?

Homework Statement NxNx[0,1) is homeomorphic to [0, 1). Find an explicit homeomorphism. (Note that N=naturals) Homework Equations A function f is a homeomorphism if: (1) f is bijective (2) f is continuous (3) f inverse is continuous The Attempt at a Solution Finding a map from...

I realized this a few hours after posting, so your response reinsured what I had come up with. Thank you. Drawing a picture of XxX with X=reals made this very clear. It's amazing how much drawing a simply illustration can help.

Homework Statement Let X be an infinite set with the cofinite topology. Show that the product topology on XxX (X cross X) is strictly finer than the cofinite topology on XxX. Homework Equations None The Attempt at a Solution So we know that a set U in X is open if X-U is finite...

Suppose we have a decimal string in base 2 (ex: 0.10111000...) then are there any of these that equal the same number in [0, 1]? I was never formally introduced to anything like this, yet I'm being asked questions involving base 2, base 3, etc. If someone could answer this, it would help me...

By Peano's space-filling curve, there exists a continuous map f: I -> I^2 whos image fills up the entire square I^2 (where I=[0, 1]). This can also be represented by gluing points of I together. Which points of I get glued together? I was looking at the proof of Peano's space-filling curve...

I think it just means that the space has that specific metric. The balls are the same as they are in the reals under the usual metric.

A ball is under the regular metric of the reals. If each singleton is open, wouldn't that make them all closed as well since the complement of each singleton is just an arbitrary union of open sets? I'm convinced that no set in this space is dense by the way, unless anyone would like to...

I havn't convinced myself that every point in S is an open set. My textbook has a definition for open sets as the following: A set G in S is is called open if for each x in G, there is an r>0 s.t. B(x, r) is in G. This is clearly not the case for single points of S, as any open ball is...

Homework Statement Let S={1/k : k=1, 2, 3, ...} and furnish S with the usual real metric. Answer the following questions about this metric space: (a) Which points are isolated in S? (b) Which sets are open and closed in S? (c) Which sets have a nonempty boundary? (d) Which sets...

Homework Statement Show that the following three conditions of a metric space imply that d(x, y)=d(y, x): (1) d(x, y)>=0 for all x, y in R (2) d(x, y)=0 iff x=y (3) d(x, y)=<d(x, z)+d(z, y) for all x, y, z in R (Essentially, we can deduce a reduced-form definition of a metric space...