Compact Definition and 309 Threads

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...

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...

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 .

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...

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...