Compact Definition and 325 Threads

Homework Statement Let E be the set of all x\in [0,1] whose decimal expansion contains only the digits 4 and 7. Is E compact? Is E perfect? Homework Equations The Attempt at a Solution My answer is: E is compact and perfect. By Heine-Borel theorem (E is compact equivalent to E...

What are the best books (upper undergraduate / beginning graduate) for stellar structure, stellar evolution, compact stars, etc. ?

Hi, I have come across the following apparent contradiction in the literature. In "Joyce D.D., Compact manifolds with special holonomy" I find on page 125 the claim that if M is a compact Ricci-flat Kaehler manifold, then the global holonomy group of M is contained in SU(m) if and only if the...

Is a single point in R compact? It seems obvious since every open cover of a single point in R can clearly have a finite subcover. However, I have a little uncertainty (i.e possible convention that says otherwise?) so just wanted to check before using it in a proof. thanks

I've only just started getting into Topology and a few examples of compactedness have me a little confused. For instace, the one in the title: how is the open interval (0,1) not compact but [0,1] is? Obivously I'm making some sort of logical mistake but the way I think about it is that there...

Someone here once said to me, via post, that "any compact spacetime must have closed timelike curves". Are there any good references out there on why that is / how that is derived? As an after thought... Isn't it true that a particle traveling in one direction in time is equivalent to its...

Homework Statement Is the following statement true: for every compact metric space X there is a constant N S.T. every subcover of X by balls of radius one has a subcover with at most N balls? Homework Equations The Attempt at a Solution I know you're meant to post your working but I really...

Hi, if I know that A is continuous and linear and A*A is compact, where A* is the adjungate, how will I show that A has to be compact?

Suppose there is an extra compact spatial dimension in addition to familiar space dimensions x, y, and z. Let us suppose that matter is some kind of 3 dimensional surface moving in these 4 spatial dimensions, 4-space, in some cyclical manner. Let us suppose that from the shape of the surface we...

Homework Statement Let C([0,1]) be the metric space of continuous functions on the interval [0,1] with distance = max of x over [0,1] of |f(x)-g(x)|. Is the ball of radius 1 centered around f(x) = 0 compact? The Attempt at a SolutionI originally thought it was but now I believe that it is...

It is a fact that if X is a compact topoloical space then a closed subspace of X is compact. Is an open subspace G of X also compact? please consider the following and note if i am wrong; proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the...

It is a fact that if X is a compact topoloical space then a closed subspace of X is compact. Is an open subspace G of X also compact? please consider the following and note if i am wrong; proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the...

There's a theorem that says any nested sequence of compact sets in Rn always has a non-empty intersection. So there is something wrong with this counterexample. I'm not able to see what's wrong: Consider the interval Un = [2-1/n, 1+1/n] for n=1, 2 and 3. Isn't the intersection of U1, U2 and...

Homework Statement [/b] I need help proving that if X is a metric space and E a subset of X is compact, then E is sequentially compact. I know I need to consider a sequence x_n in E, and I want to say that there is a point a in E and a radius r > 0 so that Br(a) [the ball of radius r with...

That's what somebody thought about on chat, yesterday (MiH). So you could go to the shop and exchange a random piece of hardware for some paper sheets with portraits of past presidents. But you could also look into the matter somewhat closer and see what the specialists recommend based on...

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...

Let M be a three dimensional Riemannian Manifold that is compact and does not have boundary. I believe manifolds that are compact and without boundary are called closed. So, my manifold M is closed. I'm interested in knowing the answers to the following questions. Under what conditions is...

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...

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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...