Compact Definition and 325 Threads

Here is the exercise: ---------- Let (X,d_{disc}) be a metric space with the discrete metric. (a) Show that X is always complete (b) When is X compact, and when is X not compact? Prove your claim. --------- Now (a) is pretty simple, but for (b) I am still not sure. Here is our definition of...

According to definition, a compact set is one where every open cover has a finite sub-cover. So let say I have C1, which is an open cover, I have C2 subset of C1 which is also an open cover. But C2 is finite. But since C2 is an open cover then there is a finite subcover C3 which is subset of...

Homework Statement A digital audio compact disc carries data, with each bit occupying 0.6 (mu)m, along a continuous spiral track from the inner circumference of the disc to the outside edge. A CD player turns the disc to carry the track counterclockwise above a lens at a constant speed of 1.3...

question on "If a set is unbounded, then it cannot be compact" Hello, I am not a mathematician so wanted to understand by picturizing and got stuck in between. While trying to understand the proof as given in Wikipedia http://en.wikipedia.org/wiki/Heine-Borel_theorem I was not sure...

A trivial, yet difficult question. How would one prove that the real numbers are not compact, only using the definition of being compact? In other words, what happens if we reduce an open cover of R to a finite cover of R? I let V be a collection of open subset that cover R Then I make the...

What would happen if we applied high potential difference to a mixture of deuterium and tritium gases in a superconducting tube?:confused: Would the electric discharge give suffecient energy and conditions for fusion to occur??:rolleyes:

Suppose K\subset\mathbb{R}. Prove that K is compact \Leftrightarrow (whatever be the indexing set I, i\in I, F{i}\subset\mathbb{R}, F_i are closed such that for all finite J\subset I, with \bigcap_{i\in J}F_i\bigcap K\neq\emptyset\Rightarrow\bigcap_{i\in I}F_i\bigcap K\neq\emptyset). I can't...

Could someone explain me how these three concepts hang together? (When is a set bounded but not closed, closed but not bounded, closed but not compact and so one?)

Hi all, you all sure know this theorem. We've it as follows: Let (P,\rho) be metric space, let K \subset P be compact and let f:\ K \rightarrow \mathbb{R} is continuous with respect to K. Then f has its maximum and minimum on K. Proof: f(K) is compact (we know from previous...

The Q: Define the distance between points \left( x_1 , y_1\right) and \left( x_2 , y_2\right) in the plane to be \left| y_1 -y_2\right| \mbox{ if }x_1 = x_2 \mbox{ and } 1+ \left| y_1 -y_2\right| \mbox{ if }x_1 \neq x_2 . Show that this is indeed a metric, and that the resulting...

EDIT: I posted this in the wrong forum, will repost in textbook questions. Please delte this (or move it). The Q: Define the distance between points \left( x_1 , y_1\right) and \left( x_2 , y_2\right) in the plane to be \left| y_1 -y_2\right| \mbox{ if }x_1 = x_2 \mbox{ and } 1+...

The question reads: Is it true that every compact subset of \mathbb{R} is the support of a continuous function? If not, can you describe the class of all compact sets in \mathbb{R} which are supports of continuous functions? Is your description valid in other topological spaces? The answer to...

This is something that I think I should already know, but I am confused. It really seems to me that the set of all real numbers, \Re should be compact. However, this would require that \Re be closed and bounded, or equivalently, that every sequence of points in \Re have a limit...

Question Let X be a compact Hausdorff space and let f:X\rightarrow X be continuous. Show that there exists a non-empty subset A \subseteq X such that f(A) = A. At the moment I am trying to show that f is a homeomorphism and maybe after that I can show that f(A) = A. But I am not sure if this...

The last paragraph of http://arxiv.org/PS_cache/physics/pdf/0006/0006039.pdf states this conclusion: See these references also: http://physics.ucr.edu/Active/Abs/abstract-13-NOV-97.html http://www.everythingimportant.org/viewtopic.php?t=79...

How can you prove that the set of orthogonal matrices are compact? I know why they are bounded but do not know why they are closed.

Hi there. I'm taking a course in analysis and I was thinking about the relation between compact sets and homeomorphism. We know that if f is an onto and one-to-one homeomorphism then it follows that for every subset K: K is compact in M <=> f(K) is compact in N Now, does this go the...

eddo's thread got me thinking: How can you tell if a specific topological space is compact? It seems like it would be hard to do just starting with the definition of compactness.

If f : U → R is continuous on U, and E ⊂ U is closed and bounded, then f attains an absolute minimum and maximum on E. How do you prove this theorem? I asked about this a while ago, but now I have a chance to redo the assignment and I need to fix my proof. I started by proving it for the 1...

:bugeye: This is a topology question. Does anybody have any suggestions on how to prove that a topological space X is COUNTABLY compact (i.e. every COUNTABLE open cover has a finite subcover), IF AND ONLY IF, EVERY NESTED SEQUENCE of closed nonempty subsets of X has a nonempty intersection...

Construct a compact set of real numbers whose limit points form a countable set.

I'd like hints only please. I have an analysis book and I could look up the proof myself but I'm trying to prove it myself as an exercise; so giving the full proof would be redundant as well as counterproductive to my own learning. X is a metric space. In this other book, K is compact iff...

Hi I'm looking for a guide to introduce muyself in the study of compact and non compact Lie algebras. Please take a minute to signal me some bibliography al the respect. Thank very much Guillom

can you list out the companies manufacturing compact pulsed microlaser source in blue/green wavelengths?

A compact car, mass of 725 kg, is moving at 100 km.hr. What is its momentum? At what velocity is the momentum of a larger car, mass 2175 kg, equal to that of the smaller car? this is what i have so far: p=725(100 km/hr) but i don't kno how to change km/hr into m/sec. And i don't know...