Recent content by Saketh

Rudin defines an open set as one which contains only interior points -- that is, points which have neighborhoods contained within the set. Prove that neighborhoods are open, given this definition. (Rudin does it for you in the theorem you mentioned, but it would be helpful for you to reproduce...

You've shown that (x-\epsilon, x+\epsilon) is a "neighborhood" of radius \epsilon about x, but you haven't shown that it's open.

I am confused by the definition of fineness on filters. Are all filters both finer and coarser than themselves?

I'm going to try to fill in the blanks. Let p be a point on the punctured sphere. Draw a curve from P to a unique point f(P) on the plane. Let q be a point on the plane. Draw a curve from q to a unique point g(Q) on the punctured sphere. Given p and q, f maps p to q, and g maps q to p...

I would really appreciate it if someone could show me how to start such a proof. I am unable to find examples of any proofs of homeomorphism on the internet, which is why I'm struggling. I don't understand how this map maps R^n to B^n. Isn't it mapping R to B? I'm sorry for not understanding...

I'm trying to show that all elements in the domain (total preimage) are open, and that all elements in the image are open. From the bijectivity of f, it follows that for every U, f(U) is open. What is wrong with this?

Thanks for correcting me again. Let me try once more. Lemma 2: If has a continuous inverse. Proof of lemma 2: Because the inverse of f is bijective, every set in the image has one corresponding open preimage in U. Since the total preimage of f consists only of open sets, and f is...

Okay, thanks for pointing out my errors. Is this correct now? Lemma 1: f is bijective. Proof of lemma 1: U and \mathbb{R}^n have the same cardinality, therefore, since we are told that f is injective, it follows that f is also bijective. Lemma 2: f has a continuous inverse. Proof of...

I just read that the punctured sphere isn't a solid ball with a point removed, as I had thought, but a shell. This suddenly makes the second problem clear. I still don't know how to prove the first one though. How can I make this intuitive explanation of the punctured sphere problem rigorous...

Homework Statement For any open set U \subset \mathbb{R}^n and any continuous and injective mapping f : U \rightarrow \mathbb{R}^n, the image f(U) is open, and f(U) is a homeomorphism. Homework Equations N/A The Attempt at a Solution I am trying to learn how to write proofs, so...

The tangent function on the interval (-\frac{\pi}{2}, \frac{\pi}{2}) seems to be a homeomorphism because it is bijective, continuous, and its inverse (arctan) is continuous. I think I could use this to prove the first thing for \mathbb{R} , right? But how do I generalize this to \mathbb{R}^n...

I'm trying to teach myself differential geometry from the internet, and I've hit a snag in proving homeomorphisms. First, show that \Re^n is homeomorphic to any open ball in \Re^n. (I'm not sure how to write the conventional "R" using Latex.) I'm trying to prove this statement, but I am...

http://www.allaboutcircuits.com/" within the DC circuits chapter.

This is a question that I'm asking myself for my own understanding, not a homework question. I realize that in most derivations of the Euler-Lagrange equations the coordinate system is assumed to be general. However, just to make sure, I want to apply the "brute force" method (as Shankar...

Ah ha! Then my intuition was correct. I wasn't sure what you call a light sonic boom, but now I know! Would this light be ultraviolet/purple? Oops, now that I think about it the title is misleading...sorry!