What is Closed: Definition and 1000 Discussions

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. S

    Half Closed Universe: Is It Finite?

    if the universe is not infinite in size is that mean a closed universe ?
  2. E

    Closed form solution of differential equation

    i have got a solution of my differential equation - consists of Bessel function and hyper-geometric function....should i call it as a closed form solution? and i would also like to know about the importance of closed form analytical solution of any problem...what is the greatness of...
  3. L

    Closed set as infinite intersection of open sets

    This is not a homework problem, just something I was thinking about. In a general metric space, is it true that every closed set can be expressed as the intersection of an infinite collection of open sets? I don't really know where to begin. Since the finite intersection of open sets is open...
  4. T

    Simple Analysis Question: Showing a Set is Closed

    Homework Statement Suppose S is a nonempty closed subset of \mathbb{R}^n, and let x \in \mathbb{R}^n be fixed. Show that A = \{d(x, y) : y \in S\} is closed. Homework Equations A set is closed if its complement is open, or if it contains all of its limit points. The Attempt at a Solution I...
  5. conquest

    Glueing together normal topological spaces at a closed subset

    Hi all! My question is the following. Suppose we have two normal topological spaces X and Y and we have a continuous map from a closed subset A of X to Y. Then we can construct another topological space by "glueing together" X and Y at A and f(A). By taking the quotient space of the disjoint...
  6. 6

    Exploring Closed Sets in Metric Spaces through Infinite Intersections

    Homework Statement Find (X,d) a metric space, and a countable collection of open sets U\subsetX for i \in Z^{+} for which \bigcap^{∞}_{i=1} U_i is not open Homework Equations A set is U subset of X is closed w.r.t X if its complement X\U ={ x\inX, x\notinU} The Attempt at a Solution Well...
  7. K

    +2uC and -2uC charges inside a closed gauss box.

    Electric Field inside a closed gauss box. Homework Statement there are +2uC and -2uC charges inside a close "Gauss" box. Which of the following statement is true? Homework Equations given option are: 1) the net electric flux through the box is zero 2) the electric field is zero...
  8. J

    Proving Sigma-Rings Are Closed under Countable Intersections

    I'm trying to prove the following and all I've got is like one line worth of proof. If we had that sigma-rings were closed under complementation, this would be easier, but we only know that if A in R and B in R, then A \ B in R and B \ A in R (symmetric difference). Is there a way to...
  9. Z

    How to get the closed form of this recurrence?

    Homework Statement Hello, This expression was derived from a polygon word problem and I need to find a closed form for it with repeated substitution (I think). T(k, n) = T(k, n-1) + (k-2)(n-1) + 1 Homework Equations The Attempt at a Solution Get a pattern like: = T(k, n-2) + (k - 2)(2n -...
  10. O

    Closed loop stabilization control - integrator circuit

    Hello all. I have a platform that is controlled by two electric motors (one for elevation, one for rotation). During the application, I would like to have the platform maintain it's current position. I'm imagining a system where you set the position manually, and then press a button that...
  11. J

    Proving C[a,b] is a Closed Subspace of L^{\infty}[a,b]

    Homework Statement Let [a,b] be a closed, bounded interval of real numbers and consider L^{\infty}[a,b]. Let X be the subspace of L^{\infty}[a,b] comprising those equivalence classes that contain a continuous function. Show that such an equivalence class contains exactly one continuous...
  12. P

    Closed Curves on the Riemann Sphere

    Is the imaginary axis considered a closed curve on the Riemann Sphere?
  13. H

    Mean Curvature at Extremum Points on a Closed Surface

    Hi, I know that the mean curvature at an extremum point where the function vanishes must be nonpositive.can this say something about the sign of the mean curvature at the farthest point on a close surface from the origin? Thank's Hedi
  14. C

    Automotive Closed loop spark advance and it's effect on abnormal combustion.

    Recently I read about Saab's experiments at closed loop spark advance, and it got me wondering. If you are controlling spark advance to maintain a constant peak pressure position of 20 deg after TDC is it possible to get into a situation where detonation or pre-ignition is possible. It seems to...
  15. C

    Convex function on closed interval?

    Homework Statement Let K be the closed interval [0,1] and consider the function f(x)=x^2. Is f convex? Is f linear? help please :/ i don't even know how to set this up to check, our teacher didn't even get to this in class yet! Homework Equations The Attempt at a Solution
  16. D

    Is W Closed in the Space C[-Pi, Pi]?

    Homework Statement W is a subset of C[-Pi,Pi] consisting of all finite linear combinations: 1,cos(nx),sin(nx) i) Show that W is a subspace of C[-Pi,Pi] ii) Is W closed in C[-Pi,Pi]. Hint from Fourier analysis: For x in [-Pi,Pi]...
  17. H

    Hydraulic Problem: Fluid backing up into a closed tube

    Hello, I am investigating a phenomenon known as "Blood reflux" and I am interested in the Hydraulic Principles I need to understand to solve a problem related to this. Here is the problem: PROBLEM: - A 20 gauge plastic tube is inserted into a vein with a 6mm diameter. The end of this...
  18. M

    Measuring the Age of the Universe: Exploring Light & a Closed Universe

    How is the age of the universe measured? Is it by the distance light has traveled since the big bang? Does that imply a closed universe?
  19. T

    Finding a closed form of the following

    Homework Statement Create a closed form for: ƒn = 14ƒn − 1 − 32ƒn − 2 + 24ƒn − 3 Homework Equations Initial conditions: ƒ(0) = 2 ƒ(1) = 5 ƒ(2) = 11 The Attempt at a Solution Because it's 3rd order, it has me confused as how to start it. I was thinking something along the...
  20. A

    Calculus II - closed function question

    there is a function: F ( x, y, z) = 2ln (xz) + sin ( xyz) − y^2 = 0. the func is defined by the closed function z=f(x,y) and provides : f(1,0)=1 we define: g(t)=f(t,1-t^6) . where t is very close to 1. I have to find g'(1)Homework Equations I tried to to do like that: find F'x and F'z and did...
  21. A

    Sets closed under complex exponentiation

    The rational (and also algebraic) elements of ℂ are closed under addition, multiplication, and rational exponentiation (the algebraic numbers, that is), but not under complex exponentiation. For instance, (-1)^i=e^{-\pi}, with is not rational, and in fact it is even transcendental. Is there any...
  22. H

    Central force field-condition for closed orbits.

    Homework Statement A particle moves in the central force field \overrightarrow{F}=-kr^{n}\hat{r} , where k is a constant, and r is the distance from the origin. For what values of n closed stable orbits are possible? Homework Equations The Attempt at a Solution I thought for...
  23. A

    Piston problem ( closed system )

    Homework Statement 1. A piston cylinder arrangement ( A = 0.25 m^2, P1= 200 KPa, V1= 0.05 m^3. is loaded with a linear spring. in the current configuration the spring exerts no force on the piston head. if the atmospheric pressure is 101 KPa, what is the mass of the piston head. Heat is now...
  24. F

    Topology Proof: AcBcX, B closed -> A'cB'

    Topology Proof: AcBcX, B closed --> A'cB' Homework Statement Prove: AcBcX, B closed --> A'cB' and where the prime denotes the set of limit points in that set X\B is the set difference Homework Equations Theorem: B is closed <--> For all b in X\B, there exists a neighborhood U...
  25. K

    How to determine resonance of an open or closed pipe?

    Homework Statement A pipe resonates at successive frequencies of 540 Hz, 450 Hz, and 350Hz. Is this an open or a closed pipe? Homework Equations L = (nλ)/2 or L = ((2n-1)/4)λ v = fλThe Attempt at a Solution The difference between the first two frequencies (540 & 450) is 90Hz, and the...
  26. A

    Closed Sets in \mathbb{C}: Showing Unclosedness by Example

    Homework Statement Show by example that an infinite union of closed sets in \mathbb{C} need not be closed. The Attempt at a Solution In \mathbb{R} I know that an infinite union of the closed sets A_{n}=[1/n,1-1/n] is open. Not sure if it works in \mathbb{C} as well.
  27. A

    Infinite intersection of open sets in C that is closed

    Homework Statement Find an infinite intersection of open sets in C that is closed. The Attempt at a Solution Consider the sets A_n = (-1/n,1/n). Since 0 in A_n for all n, 0 in \bigcap A_{n}. Here I'm a little stuck -- is the proof in R analogous to the proof in C, or do I need a...
  28. T

    Magnetic flux through a closed surface

    This is always zero, right? What if you construct a closed surface which only encompasses one of the poles of a magnet? Surely there would then be a non-zero flux as the inside of the surface would constitute a source (or sink) of magnetic field lines. I'm new to electromagnetism, so any...
  29. D

    Sealing a low-vacuum closed system

    I've been researching this for quite a while and feel somewhat exasperated, so I thought I would ask more knowledgeable folk. I need to seal a closed system for low vacuum, and I need to do it on a budget. My problem is that most of the information I have found deals with much higher vacuums...
  30. A

    Energy conversion in a closed system?

    We wrap a light, flexible cable around a thin-walled, hollow cylinder with mass M and radius R. The cylinder is attached to the axle by spokes of a negligible moment of inertia.The cylinder rotates with negligible friction about a stationary horizontal axis. We tie the free end of the cable to a...
  31. P

    Is the Set of Integers Closed in the Euclidean Plane?

    When considered as a subset of \mathbb{R}^2, \mathbb{Z} is a closed set. Proof. We will show, by definition, that \mathbb{Z} \subset \mathbb{R}^2 is closed. That is, we need to show that, if n is a limit point of \mathbb{Z}, then n \in \mathbb{Z}. I think this becomes vacuously true, since our...
  32. J

    Prove f is measurable on any closed set

    Homework Statement Prove if $f$ is measurable on R and C is any closed set, f^{-1}(C) is measurable. Homework Equations Definition of measurability, closed sets etc. The Attempt at a Solution I've been trying for a while to get this proof, but I seem to just end up stuck at the...
  33. P

    Integrate 1/z^2 Over a Closed Curve

    \int_\alpha\frac{1}{z^2}dz I can't figure out how to integrate this over a closed circle which contains the origin on its interior. I'm assuming it is equal to 2πi; is there a way to apply Cauchy's Integral Theorem? If I set f(z)=1/z then that is not analytic on the interior, so I don't see...
  34. K

    Rigorous definition of continuity on an open vs closed interval

    Let I be an open interval and f : I → ℝ is a function. How do you define "f is continuous on I" ? would the following be sufficient? : f is continuous on the open interval I=(a,b) if \stackrel{lim}{x\rightarrow}c \frac{f(x)-f(c)}{x-c} exists \forall c\in (a, b) is this correct? Also, what...
  35. C

    Q: How can you hear someone through a closed door?

    A question about the simplest of things: Based on the physics of sound, how can you hear someone through a closed door? I'm quite confused because someone once told me its was because the sound passes through the door; since its causing the air molecules to vibrate, when this vibration hits...
  36. S

    Real Analysis Question: Sequences and Closed Sets

    Homework Statement Let {xn} be a sequence of real numbers. Let E denote the set of all numbers z that have the property that there exists a subsequence {xnk} convergent to z. Show that E is closed. Homework Equations A closed set must contain all of its accumulation points. Sets with no...
  37. M

    Difference between open and closed Universes?

    What is the difference between a closed Universe and an open Universe? Please explain in layman's terms and describe what type our Universe is.
  38. A

    Proving |f(z)|≤ M in a Closed Contour C

    This problem is from Mathematical methods for physicists by Arfken, problem 6.4.7. A function f(z) is analytic within a closed contour C (and continuous on C). If f(z) ≠ 0 within C and |f(z)|≤ M on C, show that |f(z)|≤ M for all points within C. The hint is to consider w(z) = 1/f(z). I have...
  39. A

    Infinite union of closed sets that isn't closed?

    So I have to find an infinite union of closed sets that isn't closed. I've thought of something that might work: \bigcup[0,x] where 0\leq x<1. Then, \bigcup[0,x] = [0,1), right?
  40. P

    How long does it take to produce H2 in a closed beaker

    hi, I've got a problem which I cannot solve. The problem says that How long does it take to produce H2 in a closed beaker with volume 5l pressure 10Mpa and temp. 20 C with a current of 0.7A. First thing I did was to calculate the amount of H2 in that beaker. Assuming that it behaves ideally I...
  41. T

    What are the necessary conditions for a closed subset in metric spaces?

    Homework Statement If f:\mathbb{R}\to\mathbb{R} and g:\mathbb{R}\to\mathbb{R} are continuous functions show that: (a) the graph of f, \{(x,f(x)) : x\in\mathbb{R} \} is a closed subset of \mathbb{R}^2. (b) \{ (x,f(x),g(x)) : x\in \mathbb{R} \} is a closed subset of \mathbb{R}^3. The...
  42. T

    Is the Graph of a Continuous Function a Closed Set?

    Suppose f:\mathbb{R}\to \mathbb{R} is a continuous function (standard metric). Show that its graph \{ (x,f(x)) : x \in \mathbb{R} \} is a closed subset of \mathbb{R}^2 (Euclidean metric). How to show this is closed?
  43. S

    Pressure Calculation in Closed Loop PV Thermal

    Hi I am trying to design a closed loop testing system for a solar thermal system using water as the heat transfer medium. There a number of fittings and and pipework involved, the water might reach temperatures up to 80/90 deg C. I am trying to calculate what the max pressure in the...
  44. R

    Termal measurements in a closed chamber

    Hi! I'm having problem with measuring the temperature in a closed (vacuum) chamber. I'm using thermocouples (type T) connected to the object inside the chamber, but both the copper wire and the konstantan wire are then connected to copper wires before exiting the chamber. Therefore I cannot set...
  45. T

    Reviewing for Analysis Exam- can't figure out pf. about Set of Lim. Pts. being closed

    Homework Statement Theorem: Given a metric space \left(X,d\right), the set of all limit points of a subset E\subset X, denoted E' is a closed set. I have an Analysis Exam tomorrow and have been studying for quite awhile and last week, my professor gave us a list of Theorems to know the proofs...
  46. J

    (a) If A and B are disjoint closed sets in some

    (a) If A and B are disjoint closed sets in some ... Homework Statement (a) If A and B are disjoint closed sets in some metric space X, prove that they are separated. (b) Prove the same for disjoint open sets. (c) Fix p in X, ∂ >0, define A to be the set of all q in Z for which d(p,q)...
  47. S

    Plots of B•dl as a function of position along the closed path

    Homework Statement Two infinitely long current carrying wires run into the page as indicated. Consider a closed triangular path that runs from point 1 to point 2 to point 3 and back to point 1 as shown. Which of the following plots best shows B•dl as a function of position along the closed...
  48. I

    Deriving Hubble redshift in closed Universe from Maxwell equations

    Homework Statement I should derive the Hubble law redshift from Maxwell equations in closed Universe. Homework Equations The metric of closed Universe is ds^2 = dt^2 - a^2(t)\left(d\chi^2 + \sin^2 \chi d\theta^2 + \sin^2 \chi \sin^2 \theta d\phi^2\right). The Hubble law redshift: \frac...
  49. M

    Extending a uniformly cont function on an open interval to a closed interval?

    Homework Statement Show that the function f: J → ℝ is bounded if f is uniformly continuous on the bounded interval J. Homework Equations J is a bounded interval, so say J = (a,b) f is uniformly continuous on J, so \forall \epsilon > 0 there exists a \delta > 0 such that for s,t \in J...
  50. T

    Fuel pressure change in closed fuel tank

    Hi, I am measuring pressure at the bottom of a fuel tank and temp variations are giving me a tough time. I wish to clear my head so here goes my reasoning as temp increases - Tank Volume increases causing - height of fuel to decrease, pressure at the bottom of tank to decrease - Fuel...
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