Closed Definition and 1000 Threads

Prove that every closed set in $\mathbb{R}$ is the intersection of a countable collection of open sets. Let $G_n$ be a countable collection of open sets. Then we would have 2 cases either $x\in\bigcap G_n$ which is a point which is closed. Or we could $(a,b)$ in all $G_n$ but how to show that...

Homework Statement This question has two parts : (a) Let F be a finite collection of closed sets in ℝn. Prove that UF ( The union of the sets in F ) is always a closed set in ℝn. (b) Let F be a finite collection of closed intervals An = [\frac{1}{n}, 1- \frac{1}{n}] in ℝ for n =...

Homework Statement a fluid undergoes a reversible adiabatic compression from .5 MPa, .2 m3 to .05 m3 according to the law PV1.3 = constant. determine the change in enthalpy, internal energy and work transfer during the process. Homework Equations P1V11.3 = P2V21.3 dH = T dS + V dP dS = 0...

Homework Statement Let M be a metric space and A\subseteqM be any subset: Prove that if A is contained in some closed ball, then A is bounded. Homework Equations Def of closed-ball: \bar{B}R(x) = {y\inM:d(x,y)≤R} for some R>0 Def of bounded: A is bounded if \existsR>0 s.t. d(x,y)≤R...

Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed ball Homework Statement The full problem is: Let M be a metric space an A\subseteqM be any subset. Prove that the following are equivalent: a)A is bounded. b)A is contained in some closed ball c)A is contained in...

How does Gauss's divergence law work in a closed finite universe? Let's say the universe were a 4-sphere, with a single electron. How can I work out the field of the electron? If I draw a 3-sphere around the electron, then I split space into two regions. One region contains an electron, so...

Homework Statement A closed tube resonates at 440 Hz when it is 0.195 m long and 0.586 m long. Determine the speed of sound. How do I do this? Homework Equations wavelength=4L v=fw The Attempt at a Solution v= 440 (4L) I'm just not sure how to use both lengths given, can someone...

Could you please give me a hint on how to show that a set of operators with a property P is closed under addition? In other words, how one could prove that a sum of any two operators from the set still possesses this property P. The set is assumed to be infinite. Any references, comments...

If we define a set, c, to be R^2, how is it open and closed? The definitions I'm using: Open set: Open set, O, is an open set if for all points x are in O, and we can find ONE B(x,ρ) such that B(x,ρ) is less than zero. Closed set: Compliment of an open set, AKA R^n/O. This isn't a HW...

If not, does the second law of thermodynamics even apply? What role would entropy play if it is not?

I'm looking at Rao: Topology, Proposition 1.5.4, "A closed subspace of a Lindelöf space is Lindelöf." He gives a proof, which seems clear enough, using the idea that for each open cover of the subspace, there is an open cover of the superspace. But I can't yet see where he uses the fact that the...

Proving a Set is Closed (Topology) Homework Statement Let Y be an ordered set in the order topology with f,g:X\rightarrow Y continuous. Show that the set A = \{x:f(x)\leq g(x)\} is closed in X. Homework Equations The Attempt at a Solution I cannot for the life of me figure...

Hi, I was reading over a solution after working on a problem and got confused about some parts: http://nweb.math.berkeley.edu/sites/default/files/pages/f10solutions.pdf (first problem) First, how do we know that there are disjoint open sets U and V for each of the separated sets? (does...

The problem: Appreciate help on the following Hot water flows in an insulated copper pipe L long starting at temperature, T0 Need the temperature history, T(t,x). T(0.x)=0 T(t,0)=T0 Heat transfer coefficient(conductance) water to pipe is U. Pipe heat capacity per unit length is C I...

Homework Statement Determine VC for the network in Fig. 7.24 (left-hand image). Homework Equations Kirchhoff's Voltage Law: The algebraic sum of the potential rises and drops around a closed path (or closed loop) is zero. The Attempt at a Solution This is an example problem...

[EDITED] Why is the electric flux through a closed surface zero? My book based the reasoning from a uniform electric field. I understand that for a uniform electric field, the electric field doesn't diminish in magnitude as we move away from the source, and if we construct a box-shaped closed...

Give an example of: a) a closed set S\subsetℝ and a continuous function f: ℝ-->ℝ such that f(S) is not closed; b) an open set U\subsetℝ and a continuous function f: ℝ-->ℝ such that f(U) is not open Solution: a) e^x b) x^2 here's my problem, this is what was given in the...

Homework Statement I'm doing a project right now which deals with designing a closed loop recirculating system using water to cool multiple components. However, this post will only deal with the fluids portion. Below is the diagram of a very simplified system. Note that the areas marked red...

Suppose that 1<p<inf and a=(a_k) a complex sequence such that, for all x in l_p, the series (which runs from k=1 to inf) Sigma(a_k x_k) is convergent. Define T:l_p--->s by Tx=y, where y_j=Sigma(a_k x_k) (where j runs from 1 to j). I need to prove that 1) T has a closed graph (as a linear...

My book says that the total vector displacement around a closed loop is zero. Is this a general thing for every type of closed loop? If so, should this be obvious?

Hi Guys, Attached is a problem from an old exam for a Process Control and Instrumentation unit. I have tried everything I know (which isn't much, it's not the main assessable portion of the unit). Other questions similar involve giving us either the characteristic closed loop equation...

I have a pressure flow problem where I'm trying to understand the velocity profile of a fluid in an annular space between a stationary exterior cylinder and a rotating, longitudinally advancing cylinder at its center. So the boundary conditions a zero velocity at the exterior surface and a...

Homework Statement Prove: If ##X## is a (possibly infinite dimensional) locally convex space, ##L \leq X##, ##dimL < \infty ##, and ##M \leq X ## then ##L + M## is closed. Homework Equations The Attempt at a Solution ##dimL < \infty \implies L## is closed in ##X## ##L+M = \{ x+y : x\in L, y...

In general, how do you prove that a given trajectory in phase space is closed? For example, suppose the energy E of a one-dimensional system is given by E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4, where ε is a positive constant. Now, I can easily show that all phase...

]Homework Statement A closed loop of wire 7.2 x10^{}-3 m^{}2 is placed so that it is at an angle of 60degrees to a uniform magnetic field. The flux density is changing at 0.1 T/s. The emf, in V, induced in the loop of wire is A) 3.6x10^{}-4 b)3.6x10^{}-2 6.9 Homework Equations emf...

Homework Statement This question comes out of "Introduction to Topology" by Mendelson, from the section on Identification Topologies. Let D be the closed unit disc in R^2, so that the boundary, S, is the unit circle. Let C=S\times [0,1], and A=S \times \{1\} \subset C. Prove that...

I'm looking for some study materials regarding Closed time-like curves (CTCs). Be it a book, paper or anything other. It is highly accepted, but I'm particularly looking for a book that includes it and similar topics.

Purpose of the "closed box" It's often stated that GR follows from the observation that an experimenter inside a closed box in freefall could not distinguish between the box being in that circumstance, and the box being in open space away from any gravitational field. In each case, objects...

If my understanding is correct, they use shapes of things great distances apart and they compare certain properties measured to what is calculated for a closed, curved or flat universe. But my questions is if a 2-manifold is topologically homeomorphic to any 2-sphere and the same is true of...

Homework Statement Mercury is a silvery liquid at room temperature. The freezing point is -38.9 degrees celcius at atmospheric pressure and the enthalpy change when the mercury metls is 2.29 kJ/mol. Wat is the entropy change of the mercury if 50.0 g of mercury freezes at these conditions? The...

I have 3 questions concerning trying to prove open and closed sets for specific sequence spaces, they are all kind of similar and somewhat related. I thought i would put them all in one thread instead of having 3 threads. 1) Given y=(y_{n}) \in H^{∞}, N \inN and ε>0, show that the set...

Homework Statement -- Homework Equations -- The Attempt at a Solution This isn't really a proper homework question so I'll just write my problem here: I'm trying to compute a closed line integral over a triangular region. I have calculated two of the sides, but am now left...

Given the domain ℂ\[-1,1] and the function, f(z)=\frac{z}{(z-1)(z+1)}, defined on this domain, the Residue Theorem shows that for \alpha a positive parametrization of the circle of radius two centered at the origin, that: \int_{\alpha}f(z)=\int_{\alpha}\frac{z}{(z-1)(z+1)} = 2\pi i Can I...

The Friedmann–Lemaître–Robertson–Walker metric is a solution of the field equations of GR. It tells us the local behavior of spacetime, that is, g(x) at a given spacetime point x If the matter density is high enough, the curvature is positive. It is said then that the universe is closed...

Homework Statement The closed cylinder of the figure has a tight-fitting but frictionless piston of mass M. the piston is in equilibrium when the left chamber has pressure p0 and length L0 while the spring on the right is compressed by ΔL. a. What is ΔL in terms of p0, L0, A, M, and k...

Hey guy's, trying to figure out another closed form formula but this time for the sum of 1/squares of the first n consecutive integers. Or in other words: 1/(1^2) + 1/(2^2) + 1/(3^2) + 1/(4^2) +1/(...= I tried using the same technique as last time by setting up the formula based on...

So we recently began electrostatics and here you encounter Gauss' law saying that the flux integral of an electric field E over a closed surface is only dependent on the charge confined within the surface. Now for a sphere that's pretty obvious why. Because since the field gets weaker...

y(4C) ≈ 7.3 + C + \frac{C}{3^{10C}} + \frac{C}{3^{20C}} + \frac{C}{3^{30C}} Would: y(nC) ≈ 7.3 + C\sum_{n = 0}^{\infty}{\frac{1}{3^{10Cn}}} Be an acceptable answer? If not, what am I doing wrong here?

Homework Statement Let T: \ell^{2} \rightarrow \ell be defined by T(x)=x_{1},\frac{1}{2}x_{2},\frac{1}{3}x_{3},\frac{1}{4}x_{4},...} Show that the range of T is not closed The Attempt at a Solution I figure that I need to find some sequence of x_{n} \rightarrow x such that...

Homework Statement Is C^{k}[a,b] closed in C^{0}[a,b]? The Attempt at a Solution C^{k}[a,b] is obviously a subset of C^{0}[a,b]. My gut feeling says no. I thought the best way would be to construct a function f_{n}(x) which converges to f(x) and where f_{n}(x) is in C^{k}[a,b] but f(x) is...

Say we have NA blue balls and NB red balls mixed together in an urn. When we pick a ball it leaves the urn. I want to find the probability of picking n red balls in a row. The probability of picking a red ball is: NA/(NA+NB) But each time a red ball leaves the urn the probability of picking...

determine a closed expression for b(n), given b(0) = 3 b(n+1) = 2+\prod (k=0 to n) * b(k) , n >= 0 , without using recursion og or the product given by \prod. I want to start out by b(n+1) -1 = (b(n)-1)^2 for n >= 0 , but I am not sure. kind regards Maxmilian

Homework Statement A closed system containing a gas is to undergo a reversible process from an intial specific volume of 2 ft^3/lbm and a intial pressure of 100 psia. The final pressure is 500 psia. Compute the work done per unit mass. a. Pv= constant b. pv^-2 = constantHomework Equations (1)...

Homework Statement When is the magnetic flux on a section of a closed surface equal to zero? A. When the magnetic field is in the direction opposite that of the section’s area vector. B. When the magnetic field is in the direction of the section’s area vector. C. When the magnetic field...

Suppose that E is a field extension of F, and every polynomial f(x) in F[x] has a root in E. Then E is algebraically closed, i.e. every polynomial f(x) in E[x] has a root in E. I've been told that this result is really difficult to prove, but it seems really intuitive so I find that...

If u is harmonic function defined on (say) the open unit disc, then it can be continuously extended to the closed unit disc in such a way that it matches any continuous function f(θ) on unit circle, i.e. the boundary of the disc. But my understanding is the same cannot be said of holomorphic...

Homework Statement Suppose you have three closed loops (ring-like) Each one has current going through it. They are set parallel to each other and the same distance away from each from each other. Suppose also the current through the one in the middle loop and the one on the right loop move...

"Find the most economical dimensions of a closed rectangular box [. . .]" Homework Statement Find the most economical dimensions of a closed rectangular box of volume 8 cubic units if the cost of the material per square unit for (i) the top and bottom is 5, (ii) the front and back is 2 and...

Homework Statement Find a closed form for the nth partial sum, and determine whether the series converges by calculating the limit of the nth partial sum. 1. 2+2/5 + 2/25+...2/5k-1 Homework Equations The Attempt at a Solution What I did was I found out it was a geometric...

I tried very long time to show that For closed subset A,B of R^d, A+B is measurable. A little bit of hint says that it's better to show that A+B is F-simga set... It seems also difficult for me as well... Could you give some ideas for problems?