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.
Not sure if this would best fit here or under Computing and Technology, but since it has more to do with the engineering plausibility I'm putting it here for now.
I have a project which would benefit from CPUs that use a very high amount of wattage. Of course, this is the opposite of what the...
I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ...
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If ##X## and ##Y## are homeomorphic compact Hausdorff spaces, then ##C(X)## and ##C(Y)## are ##star##-isomorphic unital ##C^{*}##-algebras.
So I got the following map to work with
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$$C(h) : C(Y) \rightarrow C(X) \ : \ f \mapsto f \circ...
Given that one of the ##S_i## (let's name it ##S_{compact}##), is compact. Assume there is an open cover ##\mathcal V## of ##S_{compact}##. By definition of a compact subspace, there is a subcover ##\mathcal U## with ##n<\infty## open sets. Notice that ##\forall x\in (\bigcap_i S_i)##, ##x\in...
Prove that if ##X## is a topological space, and ##S_i \subset X## is a finite collection of compact subspaces, then their union ##S_1 \cup \cdots \cup S_n## is also compact.
##S_i \subset X## is compact ##\therefore \forall S_i, \exists## a finite open cover ##\mathcal J_i=\{U_j\}_{j\in...
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Hi. This is an idea which I just happened to think of, and I was curious if it would be at all feasible. Here's a quick sketch I drew:
The two curved mirrors should have a laser attached on one end and a video camera attached on the other. The laser would be tilted very slightly above...
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For (0,1), the collection of neighborhoods N_e of q from (0,1) is an open cover. However, there exists e>0 such that it will not have a finite sub cover. Let us take e=0.5*min{|p-q|}, where p=/=q and both are from (0,1). I am not sure if the construction of e here is right, please correct me if...
Hello everyone,
I'm trying to make a hexapod table, in the same style as this:
but I am trying to overcome the main flaw with this design - the minimum height being so high.
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Greg has kindly allowed me to post these equations which I compiled many years ago. Somehow I like them better than anything I've ever run across so maybe someone else will find them useful also.
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I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
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In the...
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Closed and Bounded Intervals are Compact ... Sohrab, Proposition 4.1.9 ... ...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of R and Continuity ... ...
I need help in order to fully understand the proof of Proposition...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of R and Continuity ... ...
I need help in order to fully understand the proof of Proposition 4.1.8 ...Proposition 4.1.8 and its proof read as follows:In the above proof by...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of R and Continuity ... ...
I need help in order to fully understand the proof of Proposition 4.1.8 ...Proposition 4.1.8 and its proof read as follows:
In the above proof by...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of ##\mathbb{R}## and Continuity ... ...
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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of \mathbb{R} and Continuity ... ...
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One of the main issues to send orbiters to (light years) faraway locations is the propulsion problem. Conventional chemical fuels cannot provide enough energy by weight to produce that much thrust.
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Hi PF!
When proving a closed ball in ##L_1[0,1]## is not compact, I came across a proof, which states it is enough to prove that the space is not sequentially compact. Counter example: consider the sequence of functions ##g_n:x \mapsto x^n##. The sequence is bounded as for all ##n\in \mathbb N...
Homework Statement
A strip of width w is a part of the plane bounded by two parallel lines at distance w. The width of a set ##X \subseteq \mathbb{R}^2## is the smallest width of a strip containing ##X##. Prove that a compact convex set of width ##1## contains a segment of length ##1## in every...
Does anyone know a model to identify Straight Compact Linear data?
I've been toying with Pearson Correlation Coefficient and am very disappointed.
https://en.wikipedia.org/wiki/Pearson_correlation_coefficient
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After some...
This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p.
Here is my...
Homework Statement
Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##.
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Hello! (Wave)
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https://lockheedmartin.com/en-us/products/compact-fusion.html
HOW COMPACT FUSION WORKS
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Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.
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Duistermaat and Kolk"s Lemma 1.8.2 and the preceding definition and notes...
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Say I have a disk in ##R^2##. How would I know if it is compact? I mean, if the disk has no boundary, then we can have a limit that is outside the set. On the other hand, a disk with a boundary contains all limit points. But this seems unsatisfactory as for the open disk, we are assuming that...
Homework Statement
Let the function ## f : R^n \setminus \{0\} \mapsto R ## be continuous satisfying ## f(\lambda x) = f(x) ## for all ## \lambda > 0 ## and nonzero ## x \in R^n ##. Prove f attains its global minimizer in its domain.
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Self studying here :D...
Let X and Y be noncompact, locally compact hausdorff spaces and let f: X--->Y be a map between them; show that this map extends to a continuous map f* : X* ---> Y* iff f is proper, where X* and Y* are the one point compactifications of X and Y.
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Homework Statement
Show that ##X## is countably compact if and only if every nested sequence ##C_1 \supset C_2 \supset ...## of closed nonempty sets of ##X## has a nonempty intersection.
Homework EquationsThe Attempt at a Solution
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I'm not sure if this is the right thread to post this, but here it is!
I'm currently trying to create a prop for one of my friends, since I found it an interesting challenge.
This is a simple drawing of what I'm trying to do:
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Magnetic mirrors were thought to be a viable solution for fusion power. But then, we found out that most of the plasma would simply escape. Is this right? So, then we realized that this method might not work. So after many years, Lockheed has come up with a similar model.
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Homework Statement
Let γs : I → Rn, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ0. Show that there exists some 0 < δ ≤ ε such that γs is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of...
Which of the operators T:C[0,1]\rightarrow C[0,1] are compact?
$$(i)\qquad Tx(t)=\sum^\infty_{k=1}x\left(\frac{1}{k}\right)\frac{t^k}{k!}$$ and
$$(ii)\qquad Tx(t)=\sum^\infty_{k=0}\frac{x(t^k)}{k!}$$
ideas for compactness of the operator:
- the image of the closed unit ball is relatively...
I'm attempting to prove that the product of two compact topological spaces is compact. My attempt at a proof runs something like this:
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I am using Lang's book on complex analysis, i am trying to reprove theorem 4.1 which is a simple theorem:
Let Compact(S \in \mathbb{C}) \iff Closed(S) \land Bounded(S)
I will show my attempt on one direction of the proof only, before even trying the other direction.
Assume S is compact
Idea...
Homework Statement
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I have found the following entry on his blog by Terence Tao about embeddings of compact manifolds into Euclidean space (Whitney, Nash). It contains the theorems and (sketches of) proofs. Since it is rather short some of you might be interested in.
My problem is that the space X= (0,1) is not sequentially compact and compact at the same time.
It is not sequentially compact:
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It is compact:
On the other...
Hello everyone,
I have been reading around that when performing the analytic continuation to Euclidean space (t\to-i\tau) one also has to continue the gauge field (A_t\to iA_4) in order to keep the gauge group compact.
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