Compact Definition and 309 Threads

In Introduction to Topology by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in ##\mathbb R^n##. This exercise succeeds another exercise where various equivalent formulations of "equivalent norms" have been given, e.g. that two norms ##\|\cdot\|_a,\|\cdot\|_b## are...

I know that there are highly directional shortwave/HF and VHF antenna designs that can be made fairly compact, but is it possible to do the same for medium wave, i.e. below 3MHz? I can’t seem to find anything helpful online about any theory that would point to a shape, let alone an actual...

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...

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I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ... I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...

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 (AND RECALL THAT ##C(X)## and ##C(Y)## are vector spaces). $$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...

Suppose ##f## is not uniformly-continuous. Then there is ##\epsilon>0## such that for any ##\delta>0##, there is ##x,y\in K## such that if ##|x-y|<\delta##, ##|f(x)-f(y)|\geq \epsilon##. Choose ##\delta=1##. Then there is a pair of real numbers which we will denote as ##x_1,y_1## such that if...

Is there a function that takes positive values only in the unit ball not including the boundary points defined by the set ##\{x^2+y^2+z^2<1\}##, and ##0## everywhere else?

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...

I want the board to use air to lift itself of with the coanda effect which is when air sticks at the bottom of somthing and proppels it upwards along with specially designed heliblades.l and I want somthing to cover tge blades and make it safe and I want it be compact like back invite future or...

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. My goal is to produce something which someone can stand on, in their home. It needs to become as flat as possible, and...

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. Actually, I have given some thought to the Fourier series and how they tie in with sampled-data...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows: In the...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand an aspect of Example 3.34 (c) on page 102 ... ... Examples 3.34 (plus some relevant definitions ...)...

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 [FONT=MathJax_AMS]R and Continuity ... ... I need help in order to fully understand the proof...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]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...

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 ... ... I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read...

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 ... ... I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read...

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. Nuclear fission provides a lot of energy by weight, but usually radiation energy doe not produce...

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...

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 I originally thought that this would be exactly what I needed, but... 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\}##. 2. Relevant results Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...

Hello! (Wave) I want to prove that if the initial data of the initial value problem for the wave equation have compact support, then at each time the solution of the equation has also compact support. Doesn't the fact that a function has compact support mean that the function is zero outside...

Helo. A problem in TOPOLOGY by Munkres states that for a ##T_1## space ##X## countable compactness is equivalent to limit point compactes(somtimes also known as Frechet compactness). Countable compactness means that every contable open covering contains a finite subcollection that covers ##X##...

https://lockheedmartin.com/en-us/products/compact-fusion.html HOW COMPACT FUSION WORKS Nuclear fusion is the process by which the sun works. Our concept will mimic that process within a compact magnetic container and release energy in a controlled fashion to produce power we can use. A reactor...

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.

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Lemma 1.8.2 ... ... Duistermaat and Kolk"s Lemma 1.8.2 and the preceding definition and notes...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.8.4 ... ... Duistermaat and Kolk"s Theorem 1.8.4 and its proof read as...

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. We are given a hint that the Weierstrass theorem states...

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. (A continuous map is...

Hello, let be ##G## a connected Lie group. I suppose##Ad(G) \subset Gl(T_{e}G)## is compact and the center ## Z(G)## of ##G## is discret (just to remember, forall ##g \in G##, ##Ad(g) = T_{e}i_{g}## with ##i_{g} : x \rightarrow gxg^{-1}##.). I saw without any proof that in those hypothesis...

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 Suppose that ##X## is not countably compact. Then there...

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: The button, battery and so on it not my problem, it's that I've no...

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. Lockheeds model is...

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: Let ##Q## and ##R## be compact, and ##Q \times R = S##. From the product topology, any open set of ##S## has to have the form ##S_{AB} = Q_A \times R_B##...

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...

Andy Resnick submitted a new PF Insights post Digital Camera Buyer's Guide: Compact Point and Shoot Continue reading the Original PF Insights Post.

Homework Statement The global topology of a ##2+1##-dimensional universe is of the form ##T^{2}\times R_{+}##, where ##T^{2}## is a two-dimensional torus and ##R_{+}## is the non-compact temporal direction. What is the Fermi energy for a system of spin-##\frac{1}{2}## particles in this...

Can someone recommend a good chemistry overview/ review? Some pdf document perhaps, not more than 200 pages or so. Notes that do not assume you are complete beginner, but you have a science (physics) background. That gives brief summaries and the highlights of all the topics that are dealt with...

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: If we define the sequence (\frac{1}{n}) we can show that it is not sequentially compact as the sequence converges to 0, but 0 \notin X. 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. I already knew that the gauge field had to be continued as well but I didn't...