Compact Definition and 325 Threads

Prove that the inverse of a continuous injective function f:A -> ℂ on a compact domain A ⊂ ℂ is also continuous. So basically because we're in ℂ, A is closed and bounded, and since f is continuous, the range of f is also bounded. Given a z ∈ A, I can pick some arbitrary δ>0 and because f is...

How to prove that SU(3) is compact?I have no idea how to do this . And What is the significance of The compactness of SU(3) on the quark model?

Ok, so basically I am trying to decide whether my mathematics is valid or if there is some subtly which I am missing: Lets say I have a 1-1 strictly increasing point-wise continuous function f: R -> R, and I want to show that the inverse function g: f(R) -> R is also point-wise continuous...

Homework Statement Let F \in \mathcal E'(\mathbf R). Prove that F \in \mathcal S'(\mathbf R). 2. The attempt at a solution Since F \in \mathcal E'(\mathbf R), there exists a continuous function f \colon \mathbf R \to \mathbf R and a nonnegative integer k such that for every \varphi \in...

Homework Statement I'm trying to prove that R with the usual topology is not compact. Homework Equations The Attempt at a Solution According to the solutions, there are two "simple" counterexamples of open coverings that do not contain finite subcoverings: (-n, n) and (n, n+2). Of course...

Does a geodesic flow on a compact surface - compact 2 dimensional Riemannian manifold without boundary - always have a geodesic that is orthogonal to the flow?

Homework Statement Show that every compact set must be closed. I am looking for a simple proof. This is supposed to be Intro Analysis proof. Relevant equations Any compact set must be bounded. The Attempt at a Solution Suppose A is not closed, so let a be an accumulation...

I have been posting on here pretty frequently; please forgive me. I have an exam coming up in functional analysis in a little over a week, and my professor is (conveniently) out of town. We proved in our class notes that if T:X\to X is a compact operator defined on a Banach space X, \lambda...

Is it true that if T: X\to Y is a compact linear operator, X and Y are normed spaces, and N is a subspace, then T|_N (the restriction of T to N) is compact? It seems like it would work, since if B is a bounded subset of N, it's also a bounded subset of X and hence its image is precompact in Y...

Is it easy to show that T: X \to Y is a compact linear operator -- i.e., that the closure of the image under T of every bounded set in X is compact in Y -- if and only if the image of the closed unit ball \overline B = \{x\in X: \|x\|\leq 1\} has compact closure in Y? One direction is (of...

So Tychonoff theorem states products of compact sets are compact in the product topology. is this true for the box topology? counterexample?

Homework Statement Let X be a compact metric space. if f:X-->X is continuous and d(f(x),f(y))<d(x,y) for all x,y in X, prove f has a fixed point. Homework Equations The Attempt at a Solution Assume f does not have a fixed point. By I problem I proved before if f is continuous with...

My quick question is this: I know it's true that any sequence in a compact metric space has a convergent subsequence (ie metric spaces are sequentially compact). Also, any arbitrary compact topological space is limit point compact, ie every (infinite) sequence has a limit point. So in general...

Hi, All: Let X be a metric space and let A be a compact subset of X, B a closed subset of X. I am trying to show this implies that d(A,B)=0. Please critique my proof: First, we define d(A,B) as inf{d(a,b): a in A, b in B}. We then show that compactness of A forces the existence of a in A...

I was reading Spivak's Calculus on Manifolds and in chapter 1, section 2, dealing with compactness of sets he mentions that it is "easy to see" that if B \subset R^m and x \in R^n then \{x\}\times B \subset R^{n+M} is compact. While it is certainly plausible, I can't quite get how to handle...

Here is my attempt at the proof: The statement is equivalent to the proposition that if X is not closed, X is not compact. If X is not closed, there exists a limit point p that does not belong to X. Every neighborhood of p contains infinitely many points of X, which tells us also that X is...

Is my claim correct?

Hello! Could anyone help me to resolve the impasse below? Th: Let G be a topological group and H subgroup of G. If H and G/H (quotient space of G by H) are compact, then G itself is compact. Proof: Since H is compact, the the natural mapping g of G onto G/H is a closed mapping...

Hi, can someone explain to me what is meant by a compact space? I don't understand the definitions on the web... my knowledge of alegbra is neglible... thanks

Homework Statement Identify the compact subsets of \mathbb{R} with topology \tau:= \{ \emptyset , \mathbb{R}\} \cup \{ (-\infty , \alpha) | \alpha \in \mathbb{R}\} . just need help on how would you actually go about finding it. I usually just find it by thinking about it. The Attempt at a...

Hi All, This is really a stupid question...I can't seem to get my head around it and it's making me depressed just thinking about it. Anyways, let's consider \mathbb{R}^{n} the set S = [0,1] is not compact (I know it is but I can't see the flaw in my argument which seems it should be...

Is the close interval A=[0,1] is compact?

Homework Statement Is A = {0} union {1/n | n \in {1,2,3,...}} compact in R? Is B = (0,1] compact in R?Homework Equations Definition of compactness, and equivalent definitions for the space R.The Attempt at a Solution A is compact, but I can't seem to find a plausible proof of it... It should...

Can you provide one to show a separable complete boundd metr. space X is not always seq. compact.

Is there any easy way to find all the compact sets of 1) the slitted plane and 2) the Moore plane? 1) defined as the topology generated by a base consisting of z\cup A where A is a disc about z with finitely many lines deleted. I believe the compact sets in this topology coincide with the...

Homework Statement Prove that E is measurable if and only if E \bigcap K is measurable for every compact set K. Homework Equations E is measurable if for each \epsilon < 0 we can find a closed set F and an open set G with F \subset E \subset G such that m*(G\F) < \epsilon. Corollary...

I'm trying to understand the proof of (ii)\Rightarrow(i) of proposition A.6.2.(1) here. The theorem says that the given definition of "locally compact" is equivalent to a simpler one when the space is Hausdorff. I found the proof quite hard to follow. After a few hours of frustration I'm down to...

hi.. how can we say a compact space automatically a locally compact? how subspace Q of rational numbers is not locally compact? am not able to understand these.. can anyone help me?

Total "absolute" curvature of a compact surface Hi! Someone could help me resolving the following problem? Let \Sigma \subset \mathbb{R}^3 be a compact surface: show that \int_{\Sigma}{|K|\mathrm{d}\nu} \ge 4\pi where K is the gaussian curvature of \Sigma. The real point is that I want...

To show that some T(x,y) = something is a metric on a set for which it is compact, we have to prove that it respects the 3 axioms of distance. right?

Greetings everyone, I'm in the process of designing a compact can crushing device for use on my college campus to crush aluminum beverage cans. The device consists of a 6ft tall x 5'' diameter steel tube mounted vertically to the inside corner of a residential style wheeled dumpster. The...

1. Let M be a compact metric space. If r>0, show that there is a finite set S in M such that every point of M is within r units of some point in S. 2. Relevant theorems & Definitions: -Every compact set is closed and bounded. -A subset S of a metric space M is sequentially compact...

Homework Statement Is the space X=\mathbb{R\backslash}\left\{ a+b\sqrt{2}\::\: a,b\in\mathbb{Q}\right\} locally compact? Homework Equations According to the baire theorem, in a locally compact hausdorf space, the intersection of dense open sets is dense. The Attempt at a Solution...

Hello, given a function f:R->R, can anyone explain what is meant when we say that "f has compact support"? Some sources seem to suggest that it means that f is non-zero only on a closed subset of R. Other sources say that f vanishes at infinity. This definition seem to contradict the...

Determine the homeomorphism classes of compact 3-manifolds obtained from D^3 by identifying finitely many pairs of disjoint disks in the boundary? I just started reading some low dimensional topology on my own and I came across this question. I have realized that based on how the...

i want to show that given any space X with the cofinite topology, the space X is sequentially compact. i have already shown that any space X with the cofinite topology is compact since any open cover has a finite subcover on X. i know that if we are dealing with metric spaces, then the...

hello friends :smile: I have a question about the compactness of the tangent bundle: assume that the manifold M is compact, does it make necessarily TM compact ? if not TM, a submanifold of TM (precisely a submanifold of vector norm equal to 1) can be compact?

Hi, I am trying to show that RP^n is the compactification of R^n. I have some , but not all I need: I have also heard the claim that every compact n-manifold is a compactification of R^n, but I cannot find a good general argument . I can see, e.g., for n=2, we can construct a...

Homework Statement This is a short question, just to check. Let X be a compact Hausdorff space, and suppose that for each x in X, there exists a neighborhood U of x and a positive integer k such that U can be imbedded in R^k. One needs to show that there exists a positive integer N such that...

Brilliant forum, wish i'd spent time browsing it years ago. In my layman's "understanding" of string theory six dimensions are compactified and usually presumed to be of very small size. My questions are: 1. Is there any mathematical or (better still) physical reason why this space does...

Homework Statement Let X be a locally compact, Hausdorff topological space. If x is an element of X and U is a neighborhood of x, find a compact neighborhood of x contained in U.Homework Equations The Attempt at a Solution Let N be a compact neighborhood of x_. The set D=Fr(N\cap\bar U) is...

Homework Statement Let K be a nonempty compact set in R2Prove that the following set is compact: S=\lbrace{p\in R^{2}:\parallel p-q\parallel\leq 1 for some q\in K}\rbrace Homework Equations I will apply Heine-Borel- i.e. a set is compact iff it is bounded and closed The Attempt at a Solution...

Homework Statement Let X be the integers with metric p(m,n)=1, except that p(n,n)=0. Show X is closed and bounded but not compact. Homework Equations I already check the metric requirement. The Attempt at a Solution I still haven't got any clue yet. Can anyone help me out?

I have 2 rechargeable 18v battery pack from cordless drill which I'm thinking of using as laptop charger and compact camera battery charger. These packs comes with 220v adaptor. Me and my 12yo son will be doing a week-long research for his school project (climate change) this Xmas break...

Homework Statement Prove that K = {(x,y) | y\geq0, x2 + y2 \leq 4 } is compact. Homework Equations The Attempt at a Solution So a set is compact iff it's closed and bounded. Closed: Should I try to show that Kc is open? So that for any point x in the compliment there is r>0...

A book I'm reading says (on page 5) that if X is compact and Hausdorff, every continuous function from X into ℂ is bounded. Why is that? I have only been able to prove it for metric spaces: Suppose that f:X→ℂ is continuous but not bounded. Choose yn such that |f(yn)|≥n. Since X is compact...

If the universe is filled with matter, and that matter causes space-time to bend, wouldn't the over-all structure be closed? Meaning, if I fly off in some random direction I would eventually "wrap around" the universe like a person moving across the surface of a sphere? If so, is this...

hello friends my question is: if we have M a compact manifold, do we have there necessarily TM compact ? thnx .

b]1. Homework Statement [/b] Let E be a normed vector space. Let (x_n) be a convergent sequence on E and x its limit. Prove that A = {x}U{x_n : n natural number} is compact. Homework Equations A is compact iff for any sequence of A, it has a cluster point, say a in A, i.e. there is a...

Homework Statement Let (X,Ʈ) be a topological space and T \subseteq X a compact subset. Show that T is compact as a subset of the space (T,Ʈ_T) where Ʈ_T is the relative topology on T. Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: T...