Compact Definition and 309 Threads

Homework Statement Suppose E1 and E2 are a pair of compact sets in Rd with E1 ⊆ E2, and let a = m(E1) and b=m(E2). Prove that for any c with a<c<b, there is a compact set E withE1 ⊆E⊆E2 and m(E) = c. Homework Equations m(E) is ofcourese referring to the outer measure of E The Attempt at a...

How is a general path called instead of being a continuous function from an interval to some topological space, where we replace the domain from an interval to a compact set, is there a name for such a function? Perhaps I should add that the compact set is also convex.

So my question here is: the divergence theorem literally states that Let \Omega be a compact subset of \mathbb{R}^3 with a piecewise smooth boundary surface S. Let \vec{F}: D \mapsto \mathbb{R}^3 a continously differentiable vector field defined on a neighborhood D of \Omega. Then...

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...

Homework Statement If α > 1, show: ∏ (1 - \frac{z}{n^α}) converges uniformly on compact subsets of ℂ. Homework Equations We say that ∏ fn converges uniformly on A if 1. ∃n0 such that fn(z) ≠ 0, ∀n ≥ n0, ∀z ∈ A. 2. {∏ fn} n=n0 to n0+0, converges uniformly on A to a non-vanishing function...

Homework Statement Show that the set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 is closed but not compact. Homework Equations set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 The Attempt at a Solution I set x = 0 and then y = 0 giving me [0,±√3] and [±√3,0] which means it is closed However, for it to...

Homework Statement Ok I created this question to check my thinking. Are the following Sets: Open, Closed, Compact, Connected Note: Apologies for bad notation. S: [0,1)∪(1,2] V: [0,1)∩(1,2] Homework Equations S: [0,1)∪(1,2] V: [0,1)∩(1,2] The Attempt at a Solution S: [0,1)∪(1,2] Closed -...

Just a simple question regarding the nature of a compact set X in a metric space S: Does X necessarily have to be infinite? That is, are compact sets necessarily infinite? Peter***EDIT*** Although I am most unsure about this it appears to me that a finite set can be compact since the set $$A...

I am reading Tom Apostol's book: Mathematical Analysis (Second Edition). I am currently studying Chapter 4: Limits and Continuity. I am having trouble in fully understanding the proof of Bolzano's Theorem (Apostol Theorem 4.32). Bolzano's Theorem and its proof reads as follows...

Information that is ordered can be compacted down to a single repeating unit i,e; 110055110055110055110055 down to just 110055 and this meant that it must have been highly ordered to be compacted down this far. So could it be that matter is also highly ordered somehow and it can be compacted...

Hey! :o If $f$ is a measurable complex function (that means that it doesn't take the values $\pm \infty$) with compact support, then for each $\epsilon >0$ there is a continuous $g$ with compact support so that $m(\{f\neq g\})<\epsilon$. Could you give me some hints how I could show that...

DISCLAIMER: This thread is a repost of another thread in the Nuclear/Particle Physics forum since I cannot delete that thread but this forum is more appropriate since it is more of a question in Nuclear Engineering than in Nuclear Physical theory. So I have been reading about the operational...

I am trying to understand the definition of compact sets (as given by Rudin) and am having a hard time with one issue. If a finite collection of open sets "covers" a set, then the set is said to be compact. The set of all reals is not compact. But we have for example: C1 = (-∞, 0) C2 = (0, +∞)...

I would like to prove [0,1], as a subset of R with the standard Euclidean topology, is compact. I do not want to use Heine Borel. I was wondering if someone could check what I've done so far. I'm having trouble wording the last part of the proof. Claim: Let \mathbb{R} have the usual...

Hi, friends! I find an interesting unproven statement in my functional analysis book saying the image of the closed unit sphere through a compact linear operator, defined on a linear variety of a Banach space ##E##, is compact if ##E## is reflexive. Do anybody know a proof of the statement...

The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by ##K_{ab} = k \delta_{ab}## for some...

Hi there. I have a project I am working on now which is kind of like a bicycle which uses powder clutches. The main goal is to make the whole thing compact by reducing the use of shafts. My initial idea of part of the mechanism is in the picture below. I was planning on...

I have a problem with this excercise. Ironically I think I can manage the part that is supposed to be hardest, here is the problem: Let (V,||\cdot||), be a normed vector-space. a), Show that if A is a closed subset of V, and C is a compact subset of V, then A+C=\{a+c| a \in A, c \in C\} is...

Homework Statement I'm trying to understand what compact sets are but I am having some trouble because I am having trouble understanding what open covers are. If someone could reword the following definitions to make them more understandable that would be great. Homework Equations...

Homework Statement I need to find an example of a set D\subseteqR is compact but f-1(D) is not. Homework Equations f-1(D) is the pre-image of f(D), not the inverse. The Attempt at a Solution I'm having trouble visualizing a function that would work for this scenario. Any clues...

Homework Statement Let p: E \rightarrow B be a covering map. If B is compact andp^{-1}(b) is finite for each b in B, then E compact. Note: This is a problem from Munkres pg 341, question 6b in section 54. The Attempt at a Solution I begin with a cover of E denote it \{U_\alpha\}. I...

Homework Statement Let M and N be two metric spaces. Let f:M \to N. Prove that a function that is locally Lipschitz on a compact subset W of a metric space M is Lipschitz on W. A similar question was asked here...

Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an orthonormal sequence consisting of the eigenvectors of $|T|=(T^*T)^{0.5}$, (${\lambda}_{n}$)...

The problem statement Let ##\mu## be a measure defined on the Borel sets of ##\mathbb R^n## such that ##\mu## is finite on the compact sets. Let ##\mathcal H## be the class of Borel sets ##E## such that: a)##\mu(E)=inf\{\mu(G), E \subset G\}##, where ##G## is open...

Homework Statement Let K \subset \mathbb{R^n} be compact and U an open subset containing K. Verify that there exists r > 0 such that B_r{u} \subset U for all u \in K . Homework Equations Every open cover of compact set has finite subcover. The Attempt at a Solution I tried...

During the time a compact disc (CD) accelerates from rest to a constant rotational speed of 477 rev/min, it rotates through an angular displacement of 0.250 rev. What is the angular acceleration of the CD? I converted 477 rev/min into 49.95 rad I converted 0.250 rev to radians which is 1.57 rad...

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

Homework Statement . Let ##(X_n,d_n)_{n \in \mathbb N}## be a sequence of metric spaces. Consider the product space ##X=\prod_{n \in \mathbb N} X_n## with the distance ##d((x_n)_{n \in \mathbb N},(y_n)_{n \in \mathbb N})=\sum_{n \in \mathbb N} \dfrac{d_n(x_n,y_n)}{n^2[1+d_n(x_n,y_n)]}##...

Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc...

Homework Statement . Let ##A \subset R^n## and suppose that for every continuous function ##f:A \to \mathbb R##, ##f(A)## is compact. Prove that ##A## is a compact set. The attempt at a solution. I've couldn't do much, I've thought of two possible ways to show this: One is to show that ##A##...

Fluorescent tube lamp (FTL), in very simple words, produces light from excitation of atoms due to bombardments of electrons. A compact fluorescent lamp (CFL) is a coiled-shaped version of FTL. http://upload.wikimedia.org/wikipedia/commons/3/31/06_Spiral_CFL_Bulb_2010-03-08_(white_back).jpg...

Homework Statement . Let ##(M,d)## be a metric space and let ##f:M \to M## be a continuous function such that ##d(f(x),f(y))>d(x,y)## for every ##x, y \in M## with ##x≠y##. Prove that ##f## has a unique fixed point The attempt at a solution. The easy part is always to prove unicity...

Let K be a compact hausdorff space, and u a borel measure on K. You are given that if A is an open set in K with A and E disjoint, we have u(A)=0. (E is a certain closed set in K) Show that for a borel set A, we have that u(AE)=u(A), where AE is the intersection. we have that...

Homework Statement . Let ##X## be a compact metric space. Prove that if ##\mathcal F \subset X## is a family of equicontinuous functions ##f:X \to Y \implies \mathcal F## is uniformly equicontinuous. The attempt at a solution. What I want to prove is that given ##\epsilon>0## there...

Homework Statement Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##. Homework Equations None The Attempt at a Solution I know that any continuous function attains both a maximum and a minimum on a compact set. I defined...

Is there any relation between compact embedding and dense embedding? Thanks in advance for your reply.

Hi everyone, :) I encountered the following question recently. :) Now I think this question is wrong. Let me give a counterexample. Take the set of real numbers with the usual Euclidean metric. Then take for example the sequence, \(\{\frac{1}{n}\}_{n=1}^{\infty}\). Then...

Hi everyone, :) Here's a question that I couldn't find the full answer. Any ideas will be greatly appreciated. I felt that the compactness of \(F_1\) and \(F_2\) could be brought into the question using the following equivalency. However all my attempts to solve the question weren't...

Litterature on "small" or "compact" dimensions? Hi! I'm reading some Kaluza-Klein theory which is an extension of normal 4D GR to a 5D spacetime in which the fifth dimension is a "small" or "compact" extra spatial dimension. I've found loads of literature on the differential geometry of...

I've prove everything except for the fact that E is bounded below. It would appear that you would need to know something about the functions taking a minimum value as well to show this using my method, so perhaps there is another way of thinking about things to show a lower bound?

Does anyone know if it is possible to construct a compact 3-manifold with no boundary and negative curvature? I ask this question in the Cosmology sub-forum because I see in various writings of cosmologists that it is often taken for granted that a negatively curved Universe must be infinite...

Homework Statement If ##A## and ##B## are compact sets in a metric space ##(M, d)##, show that ##AUB## is compact. Homework Equations A theorem and two corollaries : ##M## is compact ##⇔## every sequence in ##M## has a sub sequence that converges to a point in ##M##. Let ##A## be a subset...

Let $f$ be a continuous mapping of a compact metric space $X$ into a metric space $Y$ then $f$ is uniformly continuous on $X$. I have seen a proof in the Rudin's book but I don't quite get it , can anybody establish another proof but with more details ?

In the Principles of Mathematical analysis by Rudin we have the following theorem If $$\mathbb{K}_{\alpha}$$ is a collection of compact subsets of a metric space $$X$$ such that the intersection of every finite sub collection of $$\mathbb{K}_{\alpha}$$ is nonempty , then $$\cap\...

I know this proof is probably super easy but I'm really stuck. I don't want someone to solve it for me, I just want a hint. One way is trivial: suppose f continuous. [0,1] compact and the continuous image of a compact space is compact so f([0,1]) is compact Now the other...

I don't need basics, I know the causes how xenon flash "works" ; I need to know how do they minimise this kind of circuit in microprocessor? My question: The Flash tubes require kVs of potential difference to work. I am aware that big capacitors are required to make them work. This is okay...

Homework Statement Let S = {(a,b) : 0 < a < b < 1 } Union {R} be a base for a topology. Find subsets M_1 and M_2 which are compact in this topology but whose intersection is not compact. Homework Equations The Attempt at a Solution I'm not even sure what it means for an element of S to be...

Homework Statement The Dirichlet Prime Number Theorem indicates that if a and b are relatively prime, then the arithmetic progression A_{a,b} = \{ ...,a−2b,a−b,a,a+b,a+2b,...\} contains infinitely many prime numbers. Use this result to prove that Z in the arithmetic progression topology is not...

Theorem: Let S be a compact subset of ℝ^n. Then S is closed. Before looking at the book I wanted to come up with my own solution so here is what I've thought so far: Fix a point x in S. Let Un V_n (union of V_n's...) be an open covering of S, where V_n=B(x;n). We know that there is a...

Hi, I'm struggling to understand this concept. I think the term probably comes from functional analysis and I don't know any of the terms in that field so I'm having trouble understanding the meaning of what a compact linear operator is. I posted this in linear algebra because I'm reading...