Compact Definition and 325 Threads

Homework Statement This seems suspiciously easy, so I'd like to check my reasoning. The Attempt at a Solution I used the following theorem: If X is a Hausdorff space, then X is locally compact iff given x in X, and a neighborhood U of x, there exists a neighborhood V of x such that...

Homework Statement The title pretty much suggests everything. Let (X, d) be a metric space, and A a non empty compact subset of X. If A is compact, then there exists some a in A such that d(x, A) = d(x, a), where d(x, A) = inf{d(x, a) : a is in A}, i.e. the set {d(x, a) : a in A} has a...

Homework Statement How to define closed,, open and compact sets?Are they bounded or not? Homework Equations For example {x,y:1<x<2} The Attempt at a Solution It's is opened as all points are inner Can you please say the rule for defining the type of the set? Like for example...

Hello all,Here is my question while reading a proof. For a compact set K in a separable metrizable spce (E,\rho) and a continuous function t \mapsto f(t) , if we define D_{K} = \inf \{ t \geq 0 \; : \; f(t) \in K \} then, D_{K} \leq t if and only if \inf\{ \rho(f(q),K) : q \in...

This is not a homework question, although it may appear so from the title. So, in Munkres, Theorem 26.7. says that a product of finitely many compact spaces is compact. It is first proved for two spaces, the rest follows by induction. Now, there's a point in the proof I don't quite...

Hi everyone, I am posting this thread just to find some information. I have a science project which try to make a compact directional sound for detecting movement. Now, the electronics which my group is developing is on a small PCB board (approximately 5x5 cm) I am wondering if...

My professor proved the following: If C subset of X is a compact and A subset of C is closed then A is compact. Proof: Let U_alpha be an open cover of A. A subset of X is closed implies that U_0 = X\A is open. C is a subset of (U_0) U (U_alpha) and covers X. In particular they cover C...

Let A = {0,1,1/2,1/3,...,1/n,...}. Prove that A is a compact subset of R. Proof: Let {U_i} be an open cover for A. Therefore, there must exist a U_0 such that 0 is in U_0. Now since, U_0 is open and 1/n converges to 0, there must be infinite number of points of A in U_0. Now by the...

Homework Statement let |e-x-e-y| be a metric, x,y over R. let X=[0,infinity) be a metric space. prove that X is closed, bounded but not compact. Homework Equations The Attempt at a Solution there is no problem for me to show that X is closed and bounded. but how do I prove...

Homework Statement Show that if K is compact and F is closed, then K n F is compact. Homework Equations A subset K of R is compact if every sequence in K has a subsequence that converges to a limit that is also in K. The Attempt at a Solution I know that closed sets can be...

Homework Statement Suppose that (x_n) is a sequence in a compact metric space with the property that every convergent subsequence has the same limit x. Prove that x_n \to x as n\to \infty Homework Equations Not sure, most of the relevant issues pertain to the definitions of the space...

Hello all! I am reading through a combustion text and I am a little flustered by this notation: Am I correct in saying that the "coefficient matrices" {{\nu'_{ji}}} and {{\nu''_{ji}}} are not really "proper" matrices in the sense that there is not a vector that can multiply them that...

Take the discreet metric on an infinite set A. I understand that its closed (because it contains all of its limit points), but I don't understand why its bounded and why its not compact. Also, when they say "an infinite set A" do they mean a set that extends to infinite (say, [1,n] for...

Greetings, I'm helping out a student with her upcoming topology exam and something has be stomped. It's probably simple but I'm not seeing it at the moment. Consider a Hausdorf space (X,T). Any compact subset of X is therefore closed. The question is to prove the existence of a coarser...

Hello to all, here's another problem the answer of which I'd like to check. Let X be a non-empty infinite set, and U = {Φ} U {X\K : K is a finite subset of X} (Φ denotes the empty set) a topology on X. One needs to prove that the topological space (X, U) is connected and compact. Now, it seems...

Hi All, So all closed interval [a,b] is compact (see Theorem 2.2.1 in Real Analysis and Probability by RM Dudley) Now, Let's say I have [0,10] as my closed interval. Let My Open Cover be (0, 5) (5, 7.5) (7.5, 8.75) (8.75, 9.375) ... Essentially, The length of each open...

Let X be compactly embedded in Y. Assume also that there is a sequence f_n in X such that f_n converges to f weakly in X and strongly in Y to some function f in X. Can we say that f_n converges to f strongly in X?

Homework Statement In a first countable compact topological space, every sequence has a convergent subsequence. Homework Equations N/A The Attempt at a Solution I'm self-studying topology, so I'm mostly trying to make sure that my argument is rigorous. I understanding intuitively...

Hello, I hope I am asking this in the right area of the forums. My teacher wrote the following formula down at our last meeting, and I was wondering if it was true ( \mathcal{H} is the infinite dimension separable Hilbert space): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H}...

Hello, I hope I am asking this in the right area of the forums. I wanted to ask if the following formula is true (assuming H is infinite dimensional and separable): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx...

So when you watch those youtube videos and they put a compact fluorescent light bulb in the microwave and it starts to glow , but how come we get visible light from something that we are shooting microwaves at , it seems like conservation of energy does not hold . but then when i was...

First of all I just want to rant why is the Latex preview feature such a complete failure in Firefox? Actually it is really bad and buggy in IE too... So I am reading into Foundations of geometry by Abraham and Marsden and there is a basic topology proof that's giving me some trouble. They...

Hi, I have a question concerning the asymptotic boundary conditions on matter fields and the Riemann tensor. What is the precise relation between saying that "the matter fields go to zero at spatial infinity" and "the matter fields have compact support"? And how natural is it to state that...

Homework Statement Identify the compact subsets of \mathbb{Q} \cap [0,1] with the relative topology from \mathbb{R}. Homework Equations The Attempt at a Solution Is it all finite subsets of \mathbb{Q} \cap [0,1]? The relative topology contains single rational points in [0,1]...

Homework Statement Let (X,d) = (C[0,1], d_\infty), S_1 is the set of constant functions in B(0,1), and S_2 = \{ f \in C[0,1] | \norm{f}_\infty = 1\}. Are S_1 and S_2 compact? Homework Equations The Attempt at a Solution I am trying to use the Arzela - Ascoli theorem. For S_1, the set of...

Homework Statement Let (X,\tau) be a compact Hausdorff space, and let f : X \to X be continuous, but not surjective. Prove that there is a nonempty proper subset S \subset X such that f(S) = S. [Hint: Consider the subspaces S_n := f^{\circ n}(X) where f^{\circ n} := f \circ \cdots \circ...

1. The question is to show whether A is compact in R2 with the standard topology. A = [0,1]x{0} U {1/n, n\in Z+} x [0,1] 3. If I group the [0,1] together, I get [0,1] x {0,1/n, n \in Z+ }, and [0,1] is compact in R because of Heine Borel and {0}U{1/n} is compact since you can show that every...

Hello Physicsforums! I have a problem with the difference between complete metric space and a sequentially compact metric space. For the first one every Cauchy sequence converges inside the space, which is no problem. But for the last one "every sequence has a convergent subsequence." (-Wiki)...

Hi everyone. I wasn't really sure where to put this thread so I stuck it here, which seems the closest fit. Anyway I've been thinking about this for too long: what characterises the unit ball of a norm? Let's be specific: consider a finite-dimensional vector space, which may as well be...

I can't imagine an object more compact than a black hole for a certain radius and personally don't think it makes sense. Yet for earlier stages of the Universe, without a variable c , it gets far more compact than a black hole. Shouldn't major parts of the early universe simply collapse into...

Im working through derivations of string equations of motion from the Nambu-Goto Action and I'm stuck on something that I think must be trivial, just a math step that I can't really see how to work through. At this point I've derived the equation of motion for the closed string from the wave...

It is said on wiki* that "Maximal compact subgroups are not unique unless the group G is a semidirect product of a compact group and a contractible group, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element g in G such that...

The most recent version of the theorem, as stated by Nikonorov in 2004 Let G a connected compact semisimple Lie group, which acts almost effectively on a seven-dimensional simply-connected homogeneous space M^7=G/H. If (G/H,\rho) is a homogeneous Einstein manifold, then it is either a symmetric...

Homework Statement Let X=C[0,\pi]. Define T:\mathcal{D}(T) \to X, Tx = x" where \mathcal{D}(T) = \{ x \in X | x(0)=x(\pi)=0 \}. Show that \sigma(T) is not compact. Homework Equations None. The Attempt at a Solution Well, functions sin(Ax) and sin(-Ax), for A=0,1,2,... are in the domain...

I'm doing an experiment on Solid-State Greenhouse effects. For my setup, ideally, I require a solar simulator lamp capable of fitting inside a 1" diameter (~2.5 cm) quartz tube. The two types best capable of simulating the solar spectrum seem to be: 1. Quartz Tungsten Halogen lamps - A great...

Hi all, I've been looking over some results from functional analysis, and have a question. It seems that often times in functional analysis, when we want to show something is true, it often suffices to show it holds for the unit ball. That is, if X is a Banach space, then define b_1(X) =...

Homework Statement Let K be a compact sebset of a metric space (X, d) and let \epsilon greater than 0. Prove that there exists finitely many points x_1 x_2, ... x_n \in K such that K is a subset of the union of the \epsilon neighborhoods about x_i Homework Equations N/A The Attempt...

Homework Statement Prove that if every continuous real-valued function on a set X attains a maximum value, then X must be compact.Homework Equations None really.The Attempt at a Solution None really, not sure where to start. I know that if a space is compact, every function attains it's minimum...

Homework Statement I am trying to prove that, if X is compact and Y is closed, X+Y is closed. Both X and Y are sets of real numbers. Homework Equations The Attempt at a Solution I know that a sum of two closed sets isn't necessarily closed. So I presume the key must be the...

Is it true that every open set contains a compact set?

Homework Statement Show that S = [0,1) is not compact by giving an closed cover of S that has no finite subcover. Homework Equations The Attempt at a Solution I know that S is not compact because it is an open not a closed set even though it is bounded. But I am completely...

Homework Statement Show that if E is a closed subset of a compact set F, then E is also compact. Homework Equations I'm pretty sure you refer back to the Heine-Borel theorem to do this. "A subset of E of Rk is compact iff it is closed and bounded" The Attempt at a Solution We...

Homework Statement Show that if X is a locally compact space but not compact, then B(C_0 (X)) has no extreme points, in which B(X)=\{ x | \; ||x|| \leq 1 \} and C_0(X) = all continuous function f: X \rightarrow \mathbb{F} ( with \mathbb{F} the complex plain or the real line)...

I'm taking real analysis and struggling a bit. In class today our professor was saying something about how a function may not be continuous on a non compact set or something, but anyway, he drew the closed interval from 0 to 1 but looped one end back to the middle of the interval. __...

Homework Statement Show if K contained in R is compact, then supK and inf K both exist and are elements of K. Homework Equations The Attempt at a Solution Ok we proved a theorem stating that if K is compact that means it is bounded and closed. So if K is bounded that means...

Homework Statement Let E and F be 2 non-empty subsets of R^{n}. Define the distance between E and F as follows: d(E,F) = inf_{x\in E , y\in F} | x - y | (a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E...

I have a bone to pick with the standard proof of the closed interval in R being compact with respect to the usual topology. The proof starts out claiming that we can divide the interval in two and it is one of these two halves that is not coverable a finite subcollection of any open cover...

Is there a way to make a compact space hausdorff while preserving compactness?

I'm talking about E \times F, where E,F \subseteq \mathbb{R}^d. If you know E and F are compact, you know they're both closed and bounded. But how do you define "boundedness" - or "closed", for that matter - for a Cartesian product of subsets of Euclidean d-space? The only idea I've had is...

I can't understand why the set \mathcal{A}=\left\{\frac{1}{2^n};\,n\in\mathbb{N}\right\} is not compact, while \mathcal{A}\cup\{0\} is. I know that set is compact if and only if it's closed and bounded, so in order to make set \mathcal{A} closed, we need to include zero, as it's condesation...