Closed Definition and 1000 Threads

Homework Statement Let (X, d) be a metric space. The set Y in X , d(x; y) less than equal to r is called a closed set with radius r centred at point X. Show that a closed ball is a closed set. Homework Equations In a topological space, a set is closed if and only if it coincides...

Homework Statement Show that the intersection of two closed sets is closed. Homework Equations The Attempt at a Solution Let X and Y be closed sets i.e. X and Y are equal to their closure X_ and Y_. Then X\capY is equal to X_\capY_.

Homework Statement I am trying to prove that, if X is compact and Y is closed, X+Y is closed. Both X and Y are sets of real numbers. Homework Equations The Attempt at a Solution I know that a sum of two closed sets isn't necessarily closed. So I presume the key must be the...

By Weierstrass approximation theorem, it seems to be obvious that every continuous function has a power expansion on a closed interval, but I'm not 100% sure about this. Is this genuinely true or there're some counterexamples?

The drawing shows two fully charged capacitors (C1 = 2.00\muF, q1 = 6.00\muC; C2 = 8.00\muF, q=12.0\muC). The switch is closed, and charge flows until equilibrium is reestablished (i.e., until both capacitors have the sam voltage across their plates). Find the resulting voltage across either...

Homework Statement Show that if E is a closed subset of a compact set F, then E is also compact. Homework Equations I'm pretty sure you refer back to the Heine-Borel theorem to do this. "A subset of E of Rk is compact iff it is closed and bounded" The Attempt at a Solution We...

Suppose f:[a,b]--> R and g:[a,b]-->R. Let T={x:f(x)=g(x)} Prove that T is closed. I know that a closed set is one which contains all of its accumulation points. I know that f and g must be uniformly continuous since they have compact domains, that is, closed and bounded domains. Now T is the...

Homework Statement Let E be a nonempty subset of R, and assume that E is both open and closed. Since E is nonempty there is an element a \in E. Denote the set Na(E) = {x > 0|(a-x, a+x) \subsetE} (a) Explain why Na(E) is nonempty. (b) Prove that if x \in Na(E) then [a-x, a+x] \subset...

Homework Statement In my fluid mechanics text, it states that the Net Force due to a uniform pressure acting on a closed surface is zero or: \mathbf{F} = \int_{surface}p(\mathbf{-n})\,dA = 0 \,\,\,\,\,\,\,(1) where n is the unit normal vector and is defined as positive pointing outward from...

Homework Statement Let E and F be 2 non-empty subsets of R^{n}. Define the distance between E and F as follows: d(E,F) = inf_{x\in E , y\in F} | x - y | (a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E...

Homework Statement Consider a closed triangular box resting within a horizontal electric field of magnitude E = 7.80 & 104 N/C as shown in Figure P24.4. Calculate the electric flux through (a) the vertical rectangular surface, (b) the slanted surface, and (c) the entire surface of the box...

The main question: Let S be a subset in Rn which is both open and closed. If S is non-empty, prove that S= Rn. I am allowed to assume Rn is convex. Things I've considered and worked with: The compliment of Rn is an empty set which has no boundaries and therefore neither does Rn. Therefore...

I have been working on this problem for a few hours and am completely stuck. It seems like a simple problem to me but when I attempt it I get nowhere. The problem is: Show that \frac{1}{3}\oint\oint_{S}\vec{r} \cdot d\vec{s} = V where V is the volume enclosed by the closed surface S=...

Homework Statement i) Construct a bounded closed subset of R (reals) with exactly three limit points ii) Construct a bounded closed set E contained in R for which E' (set of limit points of E) is a countable set. Homework Equations Definition of limit point used: Let A be a subset of...

The set \{1,2,3,4,5\ldots\}...is it closed as a subset of \mathbb{R}? I'm thinking "yes," but I'm unsure of myself for some reason. (And yes, this is just the set of positive integers.

if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way

Homework Statement Show that a hyperplane in Rn is a closed set Homework Equations The Attempt at a Solution I was thinking maybe try to prove that the complement of the hyperplane is open?

I have a plastic Gatorade type bottle that I reuse. Between uses, I fill it with hot water and shake, and then empty it. Once I empty it, if I put the lid on and wait, the bottle eventually deforms and sucks in on itself. Then when I remove the lid it returns to normal. Why does it deform? Is...

This is a follow up to my previous question, as they appear that in both the Godel Metric and the Van Stockum dust Perhaps a better way to put this is, could there be a model where you had rotation (maybe around a non-symmetrical axis?) and not get these CTCs?

Thermodynamics -- Double U Tube Manometer with 2 Closed Ends Homework Statement GIVEN: Fresh water and sea water in parallel horizontal pipelines are connected to each other by a double U tube manometer. Density of sea water is 1035 kg/m^3 Density of air is 1.2 kg/m^3 Standard temp and...

Hi. An absolutely closed metric space M is such that: If N is a meric space containing M, then M is closed in N. I would like to show that an absolutely closed metric space is complete, how do I do this? I know the proof of the converse but that's no help obviously. I know intuitively...

Hey guy this is my first post here. I am looking for a little bit of help with a past paper I've been looking over. I've had a look at it and I am drawing a blank due to being off ill when this was covered in class, if anyone would be so kind as to show me how to get started on this question I...

Searching on the net for "closed loop induction" refers mostly to motors. What terms would I use to find information on the dynamics, if I can call it that, the B field that is the out-come of an induced current in this closed loop coil? I would think that the coils architecture would produce...

Homework Statement What is the line Integral of the function f = yi-xj+zk (where i,j,k, are cartesian unit vectors) around a circle with radius R centered at the origin? Homework Equations Stokes Theorem: i.e. the integreal of some function between a and b is equal to the difference in...

question: how do i find the area under f[x,y] bounded by a closed parametric curve x[t],y[t]? it doesn't look like i can use a change of variables. it seems as though double integrals only with functions where the curve is given explicitly such as y[x] or x[y].

I am building a closed loop watering system and need help determining the minimum size pipe I need for the return drain pipe. The system will pump water from a reservoir through spray nozzles into 4 containers at a rate of 300 gallons per hour. Each container will have a drain pipe in the bottom...

Hey, I am wondering if anyone can help me understand a mathematical explanation as to how they work. From what I understand, the area under a closed curve is the same, independent of the path taken. So when doing an integral you only need to take the initial and final into account. There have...

Hey, I guess I came to right sub-topic, since I knew here I may have not noticed every thing.. One sleepless night I was thinking about what would happen IF: in a closed isolated cylinder would be alomost full of H20(+some salt if required). anode and catode in sides, so what electricity...

Homework Statement A closed air column is 60.0cm long. Calculate the frequency of the forks that will cause resonance at: a) the first resonant length b) the second resonant length Note that the speed of sound is 344m/s. Homework Equations Ln = (2n - 1) * \lambda / 4 fn =...

A photon is emitted from a star in a far away galaxy. Its energy is hv = 1000 keV Its velocity is c. When it arrives at the retina, the redshift/doppler caused the photon to have an energy less than 1000keV. Where has the energy of the closed photon packet gone?

I am seeking the answer to the following question: Given a simple closed curve on a compact Riemannian surface(a compact surface with a Riemannian metric), whether there exists, in the homology class of this simple closed curve, a (single) closed curve which has the shortest length measured with...

given a Hamiltonian H=p^2 + V(x) how can you calculate the length of the closed orbits ? , i mean in gutzwiller formula you must perform a summation over the length of the closed orbits to calculate density of states g(E) but how can you know what the lenghts are ?? .. of course for Harmonic...

Homework Statement I'm given a closed loop transfer function with a gain, K=1 and a plant H(s). I need to find the gain margin of the system. Homework Equations I know how to solve this problem: a) I find the frequency where the phase is -180 degrees. b) I find the gain at that...

I have devised a simple thought experiment which leads me to an absurd conclusion and I feel I’m missing something obvious but I can't see where I’m wrong and I hope you could help point out my error. I start with an empty space initially containing two masses that are at rest relative to...

If A\subseteq B are both subsets of a topological space (X,\tau), is it true that any closed subset of A is also a closed subset of B?

im trying to understand the theory of this. is a CTC supposed to actually bring an object back to the original time? or is it supposed to make it appear that way to an outside observer? I am reading up on it from wikipedia: http://en.wikipedia.org/wiki/Closed_timelike_curve but in the beginning...

Hi, I am struggling to understand Stephen Hawking's view of the universe as a 4D closed manifold. In a recent interview, I believe he had this to say: What I don't understand is how this theory is compatible with the scientific observation that the universe is expanding? I have 2 questions: 1)...

Homework Statement A closed cylinder has total surface area equal to 600\pi . Show that the volume, Vcm3, of this cylinder is given by the formula v = 300\pi-\pi r^3 , where r cm is the radius of the cylinder. Find the maximum volume of such a cylinder. Homework Equations...

If abs x < 1 find a closed form function (i.e. f(x) = x +1) for the following series: \sum((nx)^(2n)) (reads: the series from n=1 to infinity of nx^(2n))

Homework Statement A closed loop of wire that has uniform linear density lambda is bent into the shape shown below, with dimension as indicated. Find the electric potential at point O, assuming it is zero at infinity. (see the attachment)Homework Equations V = k q /r The Attempt at a...

Hi, In young experiment, say one slit is completely closed, what observed is looks like a single band light. But why is not a diffraction pattern? I wonder whether the reason is, the slit width(don't mean distance between slits) more narrow than single slit diffraction in Young experiment?

Homework Statement A tube closed at one end and open at the other has a fundamental frequency of 242 Hz. What is the fundamental whenboth are open? Homework Equations f (open and closed) = v/4L f (open) = v/2L v sound = 343 m/s The Attempt at a Solution f1 (open and closed) =...

Homework Statement Can anyone suggest a simple example of a metric space which has a closed ball of radius, say, 1.001 which contains 100 disjoint closed balls of radius one? I've taught myself about metric spaces recently so I'm only just getting started on it really, not really sure how...

Given that f is a function from R(=real Nos) to R continuous on R AND ,A any subset of R,IS THE closure of f(A) ,a closed set??

If there's a point charge at the origin, I want to find two closed surfaces such that the flux through one of them is zero while the other is not. I know this may seem trivial but I just want to make sure I understand the question. My answer would be that to get a zero flux, the closed...

the emf is defined as the potential difference between two points \varphi(\vec{r_1})-\varphi(\vec{r_2}). ok so let's say we make a trip round a closed circular wire with a battery to keep the current flowing then r1=r2 and so no emf has been done - is this true and if so why?

Homework Statement I am having some trouble coming up with recursive and closed form expressions of different sequences. I realize helping me with this would pretty much just be giving me the answer, but if anyone could also help me with how to think of the answers that would be nice. 1) Cn =...

Hi guys! I have one question that bothers me for reeeeally long time (and it's NOT homework! :-) I found one question which goes like this "How can physics help you escape from a totally closed empty room?" Of course it means that room doesn't have any doors or windows, it would be too easy to...

The question given is: Determine the smallest subset A of Z such that 2 ε A and A is closed with respect to addition. The answer given was the set of all positive even integers, but I was thinking that the smallest subset would be the given element and the identity element (0 in this case) so...

Homework Statement (i) Give an example of a closed non-commutative binary operation on N (the set of all natural numbers). (ii) Give an example of a closed non-associative binary operation on N. The attempt at a solution This has me stumped, there must be something simple that I'm missing. I...