Closed Definition and 1000 Threads

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.
This should not be confused with a closed manifold.

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  1. D

    MHB Closed sets intersection of countable open sets

    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...
  2. STEMucator

    Closed Set Proof: Homework Statement

    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 =...
  3. J

    Reversible Adiabatic Compression: Enthaply Calc

    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...
  4. H

    Prove that if A is contained in some closed ball, then A is bounded.

    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...
  5. H

    Is A Bounded by a Closed Ball?

    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...
  6. K

    Can Gauss's Law Explain the Field of an Electron in a Closed Finite Universe?

    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...
  7. Y

    Question about resonating in a closed tube.

    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...
  8. C

    Proving Closure of Set of Operators w/ Property P Under Addition

    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...
  9. romsofia

    Understanding R^2 Open and Closed Sets

    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...
  10. F

    Is The Universe A Closed System?

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

    Closed subspace of a Lindelöf space is Lindelöf

    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...
  12. P

    Proving a Set in the Order Topology is Closed

    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...
  13. M

    Closed separated sets in disjoint open sets

    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...
  14. Z

    Closed form solution heat problem

    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...
  15. JJBladester

    Kirchhoff's Voltage Law - Closed Loop?

    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...
  16. I

    Electric flux through a closed surface

    [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...
  17. T

    Open and Closed Set with Compactness

    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...
  18. H

    Designing a closed loop fluid system

    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...
  19. C

    MHB Closed Graph Theorem: Proving T Has Closed Graph

    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...
  20. Z

    Vector displacement around a closed loop

    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?
  21. C

    Ziegler-Nichols method and closed loop characteristic equations.

    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...
  22. M

    Seeking closed form solution of Navier-Stokes for a fluid in an annular space.

    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...
  23. C

    Prove: sum of a finite dim. subspace with a subspace is closed

    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...
  24. R

    Closed trajectories in phase space

    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...
  25. R

    Emf of a Closed Loop of Wire In a Magnetic Field

    ]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...
  26. C

    Prove Cone over Unit Circle Homeomorphic to Closed Unit Disc

    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...
  27. E

    Study materials about Closed timelike curves (CTCs)

    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.
  28. R

    Can a closed box in freefall reveal the curvature of space?

    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...
  29. R

    How can they rule out a closed universe?

    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...
  30. O

    [Thermodynamics] Calculate change in entropy of closed reversible system

    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...
  31. E

    Proving Open and Closed Sets for Sequence Spaces

    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...
  32. N

    Closed Line Integral Homework - Computing a Hypotenuse

    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...
  33. P

    F has a primitive on D ⊂ ℂ ⇒ ∫f = 0 along any closed curve in D?

    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...
  34. pellman

    How do we infer a closed universe from FLRW metric?

    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...
  35. M

    Ideal Gas Law with spring, no numbers, closed container

    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...
  36. mesa

    Is There a Closed Form for the Sum of the Reciprocals of Squares?

    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...
  37. A

    Flux integral over a closed surface

    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...
  38. I

    Write a closed form expression for the approximation y(nC)

    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?
  39. S

    Showing that the range of a linear operator is not necessarily closed

    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...
  40. S

    Proving C^k[a,b] is Not Closed in C^0[a,b]

    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...
  41. A

    Closed form expression for this?

    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...
  42. M

    Determine closed expression for b(n) without recursion or product

    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
  43. M

    A closed system containing a gas is to undergo a reversible process

    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)...
  44. M

    When is the magnetic flux on a section of a closed surface equal to zero?

    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...
  45. L

    How to prove the field extension is algebraically closed

    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...
  46. L

    Why can't holomorphic functions be extended to a closed disc?

    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...
  47. M

    Current through three closed loops.

    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...
  48. S

    Find the most economical dimensions of a closed rectangular box [..]

    "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...
  49. M

    Closed Form for nth Partial Sum of a Geometric Series

    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...
  50. H

    Sum of two closed sets are measurable

    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?
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