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. tom.stoer

    The set of the real numbers is closed

    The set of the real numbers is closed. For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point (*) A) A set is closed if its complement is open; the complement...
  2. J

    Volume of a closed surface (divergence theorem)

    If exist a formula for calculate the area of a closed curve: http://en.wikipedia.org/wiki/Green%27s_theorem#Area_Calculation, so, there is a analogous for calculate the volume of a closed surface? I search but I not found...
  3. B

    Calculate pressure inside closed container

    A small container is filled with water (30ml). Next it is heated at 140°C. I need to determine the internal pressure caused by the heating process. The containers dimensions: height : 100mm Diameter: 39 mm Volume= circa 0,0203 m³ First thing I did was to look it up at steamtables. For...
  4. B

    Is there a closed form of this expression?

    Hi, (not homework/academic) Is a closed form of the following expression possible? Either way, some pointers in the right direction would be really helpful. H(s)=\sum_{n=-\infty}^\infty \frac{k^n}{k^n+a/s} Thanks, D
  5. M

    Multivariable Calculus: Max/min values inside closed interval?

    Hi! My question is: Find maximum and minimum values of the function: f(x,y) = 2x-y+x^2+y^2 when x^2+y^2 ≤ 4 I would like to solve this without using Lagrange method. I get x=-1 and y=1/2 when using partial derivative and set it equql to 0. I can see that the maximum value...
  6. L

    Closed form for geometricish series (index squared in the exponent)?

    Closed form for "geometricish" series (index squared in the exponent)? Hi all, Is there a nice closed form for the following series? \sum_{k=0}^n x^{k^2} Even a decently tight upper bound and lower bound would be nice (obviously it is bounded by the corresponding geometric series \sum...
  7. D

    Total charge density of all electrons in the closed subshell n=3, l=2

    Homework Statement Hey guys, So the title pretty much says it. I have to find the total charge density produced by all the electrons in a closed subshell where n = 3 and l = 2. The charge density produced by a single electron is (-e)|R_{32}(r)Y_{2,m}(\theta , \phi)|^{2}Homework Equations So he...
  8. M

    Thermodynamics - closed systems

    Homework Statement A cylinder fitted with a piston contains air at 1.0 Bar and 17°C. The gas is compressed according to the law PVn = const., until the pressure is 4 bar when the specific volume is found to be 28% of the initial value. Heat is then added to the air at constant pressure until...
  9. I

    How to determine whether a system is closed or not?

    I know the definition of an open/closed system but I am still confused: 1) Why isn't C a closed system? 2) I know that a simple system's state is defined by just 2 properties (when the system is in equilibrium) but how do we eliminate the wrong answers and determine the correct answer in this...
  10. M

    Intersection nested, closed sequence of intervals

    Homework Statement . Let ##\{I_n\}_{n \in \mathbb N}## be a sequence of closed nested intervals and for each ##n \in \mathbb N## let ##\alpha_n## be the length of ##I_n##. Prove that ##lim_{n \to \infty}\alpha_n## exists and prove that if ##L=lim_{n \to \infty}\alpha_n>0##, then ##\bigcap_{n...
  11. L

    Compactness of a Closed Ball in C([0,1])

    Homework Statement Show that the closed ball in ##C([0,1])## of center ##0## and radius ##1## is not compact.2. The attempt at a solution I was given a hint, to look at the sequence of continuous functions ##f_n(x) = x^n## on the closed ball in ##C([0,1])##. Why is that sequence continuous...
  12. R

    Cusps in the evolution of closed strings

    Homework Statement This is problem 7.7 in Zwiebach's book, 2ed ed. In (b) he want us to show that near the cusp, ##y\sim x^{2/3}.## In (d), Check that the period of the motion of the closed string is ##\sigma_1/4c##. How many cusps are formed during a period? Homework Equations (b)...
  13. A

    Real Analysis: closed sets and limit points

    For the following example:(if possible give example or just state impossible 1) a bounded subset A of R for which sup A is not a limit point of A. An example is (0,1) union {7}. will this work? 2) a finite subset A of R that is not closed I think it is not possible. Please give some hints...
  14. 1

    Finding closed form of sequence.

    Homework Statement {U_0 = 9, U_1 = -3} U_(n+2) = -(5/4) U_(n+1) + (3/8) U_(n) Homework Equations The Attempt at a Solution First step was to attempt to find the common difference by trying to find the 3rd term: U_(2) = -(5/4) u_(1) + 3/8 U_(0) = -(57/8) This does not...
  15. J

    Thermodynamics first law closed system

    1) A rigid walled tank 5m3 contains helium at 10 bar. The cylinder is heated from 10 °C to 50 ° C. What is the work done during the heating cycle.
  16. C

    Can a closed universe expand forever?

    Most say that a closed universe would collapse. Recent observations show that the universe will probably keep on expanding forever. Does this mean we cannot live in a closed universe? Or, is there a loop hole which allows a closed universe to expand forever?
  17. P

    Solving Work with Constant Force and Gravity on Mass in a Vertical Circle

    So I have a constant force F acting tangentially on a mass m in a vertical circle around a loop of radius r. The mass starts from rest at the very top of the loop. The only other force is gravity, that is m*g Now I did Work=Change in energy with a system that is comprised of both the...
  18. P

    Does there exist a transformation between a line and a closed loop ?

    Dear All: For a line structure(say a very long atomic chain) and a closed loop structure( connect the head and tail of this atomic chain). Does there exists a transformation between these two structures? For example if we want to study the vibration mode of these two cases. If we already know...
  19. M

    Sum of two closed subspaces in a Banach space

    Homework Statement . Let ##E## be a Banach space and let ##S,T \subset E## two closed subspaces. Prove that if dim## T< \infty##, then ##S+T## is also closed. The attempt at a solution. To prove that ##S+T## is closed I have to show that if ##x## is a limit point of ##S+T##, then ##x \in...
  20. C

    Thermodynamics 1st law question - closed system

    Homework Statement 1kg of helium in a closed system has 40kj added by heating & 100kj removed by work during a reversible polytropic process. The initial temp of the gas is 300K & the initial pressure is 100kN/m2 Q. find the final temp of the gas & pressure Homework Equations...
  21. D

    Pressure increase in a closed system

    Hello All, I work in the refrigeration industry, and I'm trying to put a hard number on a hypothetical situation. The situation is a lapse in SOP and liquid gets trapped in a line. For this hypothetical situation the line is 100% full. What would the increase in pressure be per degree? I...
  22. X

    Transformer equivalent circuit open and closed circuit test

    I see the open and closed circuit test on the transformer section I don't understand why we doing that(open and closed circuit test) can anyone explain and give some example?
  23. Q

    Open Loop Representation of Closed Loop System

    I remember having studied that closed loop systems can be represented by open loop systems. But that seems weird..if it were possible for both the types of systems to have the same transfer function, why would they behave differently?
  24. M

    How vector area of a closed surface is zero?

    how vector area of a closed surface is zero?
  25. S

    Thermodynamics Closed System Energy Balance

    Hi everyone, I am currently reviewing for a thermodynamics exam and have come across a difficult problem while studying. Here it is: An insulated tank containing helium, a monatomic gas, at P1t = 1000kPa and T1 = 800K is connected to an initially evacuated insulated piston-cylinder device...
  26. M

    Is the Intersection of Open Sets Always Open?

    Prove that for any collection {Oα} of open subsets of ℝ, \bigcap Oα is open. I did the following for the union, but I don't see where to go with the intersection of a set. Here's what I have so far: Suppose Oα is an open set for each x \ni A. Let O= \bigcap Oα. Consider an arbitrary...
  27. B

    Calculate pressure inside closed container

    Homework Statement For the design of a reactor I need to calculate the pressure inside the closed reactor. Reactor volume = 75,36 ml Mixture inside the reactor = 30 ml H[2]SO[4] (5%) Temperature = 150 °C The Attempt at a Solution I was thinking I will need the steam tables for water -...
  28. Y

    Finding the Closed Form of a Power Series

    Homework Statement Using that \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n for |x|<1 and that f'(x) =\sum_{n=0}^{\infty} (n+1)a_{n+1}(x-x_0)^n , write \sum_{n=0}^{\infty} n^2x^n in closed form. Homework Equations The Attempt at a Solution In this series, a_n = n^2 and x_0 = 0 ...
  29. T

    Sealing a 6 torr closed system vacuum

    I wish to design a two chamber stainless steel vacuum system where the two vacuum chambers are connected via a flexible tube with a valve. The system needs to be able to be pumped down to 6 torr and must be able to maintain that vacuum for at least a year without pumping. The size of the...
  30. T

    Comparing Flux of a Closed Cylinder

    Homework Statement A closed cylinder consists of two circular end caps and a curved side. A negatively charged particle is placed outside the left end of the cylinder, on its axis. Compare the flux through the left end of the cylinder with that through the right. 2. The attempt at a...
  31. S

    Can Matter Be Transferred Between Universes?

    i wish to know if matter can be transferred from one universe to other universe? does our universe have valve like thing to let the matter in or out of the system?
  32. L

    Theroem of closed sets containing limit

    Homework Statement Theorem: Let S be a subset of the metric space E. Then S is closed iff whenever p1,p2,p3,... is a sequence of points of S that is convergent in E, we have lim n→∞ p_n ∈ S. Homework Equations The Attempt at a Solution I am having trouble understand the "if"...
  33. M

    Every closed ball in X is complete, then X is complete

    Homework Statement . Prove that if every closed ball in a metric space X is a complete subspace, then X is complete. The attempt at a solution. Let ##\{x_n\}## be a cauchy sequence in X. Then, for ##ε=1##, ##\exists## ##n_0 \in \mathbb N##: ##\forall## ##m≥n_0##,##n≥n_0##...
  34. R

    Pressure Rise in Closed Tank with 2 Inputs

    Supposing you had a cylindrical closed tank inside the body of a submarine; the top of the tank is at 200m depth. The tank has height 2.5m and internal volume 1.8m^3. A section of pipework is connected between the top surface of the submersible (level with the top of the tank) and to the...
  35. R

    Draining Closed Tank: Calculating Water Volume at 200m Depth

    Supposing you had a tank mounted to the outside of a submersible vehicle. The tank lid is left open, and the vessel that taken down to 200m sea water depth, to top of tank. The tank lid is now closed at this depth; it is assumed that the tank is completely full, with no air gap/bubble on...
  36. L

    Draw a Closed Plane Curve w/ Positive Curvature

    Homework Statement How can I draw a closed plane curve with positive curvature that is not convex The Attempt at a Solution I was thinking drawing it like a banana but more curved, will that do?
  37. M

    CSTR vs. BSTR system open or closed

    Hello, I was wondering, is it the case that a CSTR is an open system because there is flow passing in and out of the tank, thus passing the control volume, so it's open. Similarly for a BSTR, the fact that there is no flow means that it is closed? I mean in that sense, a beaker with a...
  38. T

    Proving the Even Degree Property of Vertices in Closed Trails

    Homework Statement All vertices in a closed trail have even degree. Homework Equations The Attempt at a Solution Intuitively, I know this statement is true, but I can't seem to see a clear way to show it. I know that a closed trail is a path that connects vertices, so one would follow an...
  39. A

    The pure-point subspace of a Hilbert space is closed

    (All that follows assumes we are talking about a self-adjoint operator A on a Hilbert space \mathscr H.) The first volume of Reed-Simon defines \mathscr H_{\rm pp} = \left\{ \psi \in \mathscr H: \mu_\psi \text{ is pure point} \right\}. The book seems to take for granted that \mathscr H_{\rm...
  40. B

    Why can't we whistle in a closed space?

    Try whistling in a small cup or with your palms around your mouth (with some space inside, but try to cover the holes between your fingers). It seems much harder, if not impossible. What could be the reason? If you try talking, it will sound muffled but at least you can produce and hear the...
  41. F

    Does Bernoulli's Principle Predict Pressure Drop in Constricted Pipe Flow?

    Hello everyone, Say I have a closed loop comprised of a pump and some piping which connects the inlet of the pump to the outlet of the pump. All of the piping has radius "a," except for a small section, which constricts to radius "b" for a small portion of the line (the inlet and outlet...
  42. S

    Show that limit set of dynamical system is closed

    Homework Statement Define the w-limit set (omega) of a point. Show that w(x) is closed. Homework Equations The Attempt at a Solution The definition of a limit set is the set of points to which there exists a sequence t_n→∞ such that \phi(t_n,x) → y The second question. I was...
  43. C

    Mass Flow at the Mid-Plane in a closed container

    Hi there. I am designing a type cilindrical can full of a fluid, with a temperature difference between the top and bottom. Now, after the simulation of the free convection phenomenom in COMSOL, I wanted to understand the effect of varying the temperatures, to the "fluid movement" (fluid...
  44. J

    How to calculate mass of closed Universe?

    How does one integrate the mass density over a closed Universe (a 3-sphere?) to obtain the total mass of that Universe? Is this the correct integral? M = R(t)^3 \rho\int_0^1 4 \pi r^2 \frac{dr}{\sqrt{1-r^2}} where R(t) is the radius of the Universe at cosmological time t. By making the...
  45. NATURE.M

    Can the Entropy of an Isolated System Decrease?

    So in my physics textbook, the 2nd law of thermodynamics stated in terms of entropy reads "the entropy of a closed system can never decrease." Now, shouldn't it indicate the entropy of an isolated system can never decrease. All other sources I've looked at note an isolated system, as well...
  46. W

    Weird ways of doing closed loop integrals

    I was looking at an example where it was evaluating a closed loop integral of a vector field around a triangle (0,0) (1,1) (2,0) by using greens theorem. This example was in the green's theorem section of the book so green's theorem must be used. Anyways the double integral was set up as follows...
  47. R136a1

    Is Every Closed Subset of ##\mathbb{R}^2## the Boundary of Some Set?

    I'm wondering if the following is true: Every closed subset of ##\mathbb{R}^2## is the boundary of some set of ##\mathbb{R}^2##. It seems false to me, does anybody know a good counterexample?
  48. O

    Closed integral problem: field in the plane of a magnetic dipole

    I have read about Biot-Savarts law but I have no idea how to solve it when the curve is dependent of some variable. My books in mathematics don't help, nor my books in physics. You are welcome to give me a link where I can read about this, but first let me know if I have got it right this far...
  49. A

    Writing Fourier Series for Open and Closed Intervals

    In a rigorous mathematical course I am talking, it seems to make a difference when I am given a function f and need to write its Fourier series, whether it is defined on [0,2∏] or [0,2∏). What difference does it make for my series whether it is an open or a closed interval?
  50. Z

    What are closed time-like curves and how are they related to General Relativity?

    I understand that GR allows for a method of time travel using closed time like curves (CTC)s. anyway i have a few question about this, first of is there some sort of (relativaly short) equation that discribes this. So my second question is based of a something i read in this thesis paper...
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