Compact Definition and 309 Threads

Homework Statement Let E be compact and nonempty. Prove that E is bounded and that sup E and inf E both belong to E. Homework Equations The Attempt at a Solution E is compact, so for every family{G_{\alpha}}_{\alpha\in}A of open sets such that E\subset\cup_{\alpha\in}AG_{\alpha}...

I'm stuck ... Ive proved the intersection of any number of closed sets is closed ... and Let S = { A_a : a Element of I } be an collection of compact sets...then by heine Borel Theorem ...Each A_a in S is closed...so this part is done now I just have to show the intersection is bounded...

This may be a stupid question, but I just confused myself on compactness. For some reason I can't convince myself that ANY subset of a compact set isn't compact in general; just closed subsets. Suppose K is a compact set and F \subset K. Then if (V_{\alpha}) is an open cover of K, K \subset...

Homework Statement Let X be a metric space and let K be any non-empty compact subset of X, and let x be an element of X. Prove that there is a point y is an element of K st d(x,y) leq d(x,k) for every k an element of K. Homework Equations triangle inequality The Attempt at a...

I'd like to show that [a,b] is sequentially compact. So I pick a sequence in [a,b] , say (xn). case 1:range(xn) is finite Then one term, say c is repeated infinitely often. Now we choose the subsequence that has infinitely many similar terms c. It converges to c. case 2: range of (xn) is...

Is there any inclusion relationship between completely regular Hausdorff space and compact Haudorff space? What is the example to show their inclusion relationship? Thanks.

suppose f:R^m -> R^n is a map such that for any compact set K in R^n, the preimage set f^(-1) (K)={x in R^m: f(x) in K} is compact, is f necessary continuous? justify. The answer is no. given a counterexample, function f:R->R f(x):= log/x/ if x is not equal to 0 f(x):= 0 if x=0...

Is "con't fn maps compact sets to compact sets" converse true? The question is here, Suppose that the image of the set S under the continuous map f: s belongs to R^n ->R is compact, does it follow that the set S is compact? Justify your ans. I already know how to prove the original thm, it...

I was reading in a book, says \mu is a measure with compact support K in C, meaning \mu(U)=0 for U\cap K=0.. Is \mu(K) assumed to be finite in this case? It doesn't say in the book, but they make a statement which is true if that's so. Is there usually some assumption about measures being...

Homework Statement Which subset of R are both sequentially compact and connected? Homework Equations The Attempt at a Solution The connected subsets of R are the empty set, points, and intervals. The subsets of R that are compact are closed and bounded. Thus, the subsets of...

Hi everyone. I feel like I'm really close to the answer on this one, but just out of reach :) I hope someone can give me some pointers Homework Statement Let A1 \supseteq A2 \supseteq A3 \supseteq \ldots be a sequence of compact, nonempty subsets of a metric space (X, d). Show that...

Let K\subseteq\mathbb R is a compact set of positive Lebesque measure. Prove that the set K+K=\{a+b\,|\,a,b\in K\} has nonempty interior.

Homework Statement X and Y are compact sets in R^n that are disjoint. Then there must be positive distance between the elements of these sets. Homework Equations The Attempt at a Solution since X and Y are compact , X X Y is compact. Then, for the distance function d(x in X, y...

y is a correspondence of x. X is compact. Can somebody give me an example where y is compacted valued, but the graph(x,y) is not compact. A graph will be highly appreciated.

I am struggling to prove the following: Let E be a compact nonempty subset of R^k and let delta = {d(x,y): x,y in E}. Show E contains points x_0,y_0 such that d(x_0,y_0)=delta.

Homework Statement Prove that every compact set is bounded. Homework Equations The usual compactness stuff - a compact set in a metric space X is one that, for every open cover, there is a finite subcover. The Attempt at a Solution I'm really hesitant about this question because my...

Homework Statement I'm reading the proof of a theorem and the author claims w/o justification that a weakly lower semi-continuous function (w.l.s.c.) f:C-->R attains its min on the convex weakly compact subset C of a normed space E. At first I though I saw why: Let a be the inf of f on C and...

[SOLVED] Topology: Nested, Compact, Connected Sets 1. Assumptions: X is a Hausdorff space. {K_n} is a family of nested, compact, nonempty, connected sets. Two parts: Show the intersection of all K_n is nonempty and connected. That the intersection is nonempty: I modeled my proof after the...

Homework Statement Is C^n or R^n compact? The Attempt at a Solution They are not bounded so can't be compact.

Homework Statement Is the general linear group over the complex numbers compact?The Attempt at a Solution I have a feeling it is not. It is not bounded.

Homework Statement Okay, so this is a three-part question, and I need some help with it. 1. I need to show that the function f(x) = e^{-1/x^{2}}, x > 0 and 0 otherwise is infinitely differentiable at x = 0. 2. I need to find a function from R to [0,1] that's 0 for x \leq 0 and 1 for x...

Homework Statement Suppose that fk : X to Y are continuous and converge to f uniformly on every compact subset of the metric space X. Show that f is continuous. (fk is f sub k) Homework Equations Theorem from p. 150 of Rudin, 3rd ed: If {fn} is a sequence of continuous functions on E...

1. If set A is compact, show that f(A) is compact. Is the converse true? 2. If set A is connected, show that f(A) is connected. Is the converse true? 3. If set B is closed, show that B inverse is closed. Any help with any or all of these three would be greatly appreciated. Stumped!

Homework Statement Let X be a topological space. Let A be compact in X. Let B be contained in A. Let B also be closed in X. Is it always true that B is compact in X? Homework Equations The Attempt at a Solution

Homework Statement The inner and outer radii of a compact disc are 25 mm and 58 mm. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25m/s. The maximum playing time of a CD is 74.0 min. What is the average angular acceleration of a maximum-duration...

[SOLVED] Inverse Image of a Compact Set -- Bounded? Problem: Let f : X → Y be a continuous function, K ⊂ Y - compact set. Is it true that f^{-1}(K)– the inverse image of a compact set– is bounded? Prove or provide counterexample. Questions Generated: 1. Why does compactness matter? (I...

"Artificial atoms" used to improve compact discs? This should be very entertaining for all of you. If you visit http://machinadynamica.com/ you will find many interesting devices. Some of them are complete bunk (for example, the "Clever Little Clock", the "Brilliant Pebbles", and the...

Homework Statement How would I prove that if X is locally compact and a subset of X, V, is compact, then there is an open set G with V \subset G and closure(G) compact?EDIT: X is also Hausdorff (which with local compactness implies that it is regular) if that matters Homework Equations The...

The question asks to prove directly that the closed interval is covering compact - U= an open covering of the closed set [a,b] I started by taking C=the set of elements in the interval that finitely many members of U cover. Now I need to somehow use the least upper bound theorem to show...

If f from R to R is continuous, does it then follow that the pre-image of the closed unit interval [0,1] is compact? -At first I thought of a counterexample like f=sinx but it seems that its range is not R. So will the answer be yes? And how can we prove it? Will the preimage have to be...

Homework Statement If X is a metric space such that every infinite subset has a limit point, then prove that X is compact. Homework Equations Hint from Rudin: X is separable and has a countable base. So, it has countable subcover {Gn} , n=1,2,3... Now, assume that no finite sub...

http://arstechnica.com/news.ars/post/20070811-iphone-bill-is-surprisingly-xbox-huge-lol.html

Hallo everybody! Is there anybody dealing with CMS stuff? Let's share infos here, and let's discuss the stuff related to Compact Muon Solenoid experiment simulations. I am a student, and have to work on H-->2mu ee- (Higgs to muon+ muon- electron positron) Plz, leave here any related...

It hits me every time that I replace one in the house during the nine months of the year that we run the heater. I would imagine that the energy required to make one is significantly higher than for an incandescent bulb. And they don't seem to last as long as they used to. It makes me wonder...

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.