In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (i.e., containing all its limit points) and bounded (i.e., having all its points lie within some fixed distance of each other).
Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.
One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.
The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded.
Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space.
For instance, some of the numbers in the sequence 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … accumulate to 0 (while others accumulate to 1).
The same set of points would not accumulate to any point of the open unit interval (0, 1); so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded.
In particular, the sequence of points 0, 1, 2, 3, …, which is not bounded, has no subsequence that converges to any real number.
The concept of a compact space was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem to spaces of functions, rather than geometrical points. Applications of compactness to classical analysis, such as the Arzelà–Ascoli theorem and the Peano existence theorem are of this kind. Following the initial introduction of the concept, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces. In general topological spaces, however, different notions of compactness are not necessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness, is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character.
The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well.
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...
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...
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...
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...
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...
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:
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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:
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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...
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...
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...