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

    Complex Analysis - Proving a bijection on a closed disk

    Homework Statement For each w \in \mathbb{C} define the function \phi_w on the open set \mathbb{C}\backslash \{\bar{w}^{-1}\} by \phi_w (z) = \frac{w - z}{1 - \bar{w}z}, for z \in \mathbb{C}\backslash \{\bar{w}^{-1}\} \back. Prove that \phi_w : \bar{D} \mapsto \bar{D} is a...
  2. C

    Finite intersection of closed sets is not necessarily closed

    Hi everyone, I'm reading Rudin's Analysis and in the topology section, he implies that the finite intersection of closed sets is not necessarily closed. (pg. 34) Can someone give an example of this? I can't seem to find one.
  3. S

    Could open tube and closed (at one end) tube produce the same frequency?

    Homework Statement Is it possible for a flute (tube open at both ends) 72 cm long and an oboe (tube open at one end) 64.8 cm long to produce the same note? Prove your answer. Homework Equations v=f\lambda L=(n/2)\lambda (tube open at both ends) L=((2n-1)/4)\lambda (tube open at...
  4. A

    Maximum Number of Closed Curves with zero Line Integral

    Hi All, I have been battling with this question for a while. Given a conservative vector field, we know that there are infinitely many closed paths where the line integral evaluated is zero. In fact this is the requirement for a conservative vector field: Every line integral of any closed...
  5. H

    Closed form expression for the partition function Z using the Canonical Ensemble

    Homework Statement I'm looking for a closed form expression for the partition function Z using the Canonical Ensemble Homework Equations epsilon_j - epsilon_j-1 = delta e Z = Sum notation(j=0...N) e^(-beta*j*delta e) beta = 1/(k_B*T) t = (k_B*T)/delta e N is the number of excited...
  6. S

    Thermodynamics - closed system

    Homework Statement 31.47mol of copper at 273 kelvin put inside an isolated cup along with 1 mol of water vapors at 373 Kelvin. (pressure is constant at 1 atm). ALL of the water condensed. given parameters: Cp(Cu(solid)) = 24.44 J/mol Cp(H20(gas) = 33.58 J/mol Cp(H20(liquid) = 73.35...
  7. H

    Prove a set is closed and bounded but not compact in metric space

    Homework Statement Let X be the integers with metric p(m,n)=1, except that p(n,n)=0. Show X is closed and bounded but not compact. Homework Equations I already check the metric requirement. The Attempt at a Solution I still haven't got any clue yet. Can anyone help me out?
  8. G

    Open or Closed: Analyzing the Intersection of Rationals and an Interval

    Homework Statement Given the set S = the intersection of the rationals and the interval [0, 1], is S open or closed? Homework Equations Definition of open: for all elements of S, there exists epsilon > 0 such that the neighborhood (x, delta) is a subset of S. The Attempt at a Solution...
  9. C

    Proof of Closed Sets: Cluster Points & Int. Pts

    [b]1. Prove that a set is closed if and only if it contains all of its cluster points. [b]2. Can I use part of the Lemma here that states: Every interior point of A is a Cluster point. Also what exactly is the definition of a closed set other than a set is closed if its compliment is...
  10. F

    Find Closed Form of Differential Equation: y''' - y = 0

    Hi, everyone, I need some help with the following: Homework Statement Given is, that the following power series: \sum_{n=0}^{\infty} \frac{x^{3n}}{(3n)!} is the solution to the following differential equation: y''' - y = 0. Find the closed form of the series. Homework Equations...
  11. M

    Why derivative of a closed real function is continous on open interval

    consider a real function f(x) where x\in[a,b] Why f'(x),\; x\in(a,b)
  12. M

    Limit of a sequence in a closed interval is in that interval

    Homework Statement Suppose [a,b] is a closed interval (on R), and {xn}n>=1 is a sequence such that a) xn belongs to [a,b] b) lim as n--> infinity xn = x exists prove x belongs to [a,b] Homework Equations The Attempt at a Solution Well since any sequence is bounded, then...
  13. R

    Is every orientation form on a compact smooth manifold closed?

    Is every orientation form w on a compact smooth manifold closed?(i.e. dw=0)
  14. A

    Proof of Zero Value for Vector Field Integral on Closed Surface

    What is the value of a surface integral over a closed, continuous surface of a vector field of vectors normal to the surface? The integral of ndS over S. I believe the answer is zero. Can someone direct me to a proof for an aribitrary closed surface?
  15. nomadreid

    Closed and open strings in M-theory

    Given my limited background in Physics, I am restricted to the popular literature on M-theory. After going through this literature (including articles from the very helpful "annotated list of useful literature for String Theory" posted in this rubric of the Forum), I am still puzzled by the...
  16. L

    Calculating Deflagratio Pressure in a Closed Vessel

    Forgive me because I'm not a math or physics wiz but I'm at a dead end trying to calculate the pressure generated by a deflagration inside a cylinder. I have been searching for a formula but I've come up with nothing. Here is an example scenario: I have a cylinder with a height of 9cm and...
  17. I

    Does space-time form a closed manifold around a black hole?

    Mass can curve space-time. Is it possible that space-time around a black hole is so badly curved that it forms a closed 4D manifold?
  18. H

    Proving Distance Between a Point and a Closed Set

    Homework Statement Prove that a point,a, not belonging to the closed set B has a non-zero distance from B. I.e that dist(a,B)=inf(y in a) ||a-y||>0 Homework Equations I have no idea how to start this. It is only worth a few marks and I have been told it is fairly easy but I have always...
  19. N

    Help me Understand Closed Under Addition and Closed Under Multiplication

    Help me Understand "Closed Under Addition" and "Closed Under Multiplication" Linear Algebra...matrices...etc Examples would be great. Thanks.
  20. radou

    Closed continuous surjective map and Hausdorff space

    Homework Statement Here's a nice one. I hope it's correct. Let p : X --> Y be a closed, continuous and surjective map such that p^-1({y}) is compact for every y in Y. If X is Hausdorff, so is Y. The Attempt at a Solution Let y1 and y2 in Y. p^-1({y1}) are then p^-1({y2}) disjoint...
  21. radou

    Closed continuous surjective map and normal spaces

    Homework Statement Let p : X --> Y be a closed, continuous and surjective map. Show that if X is normal, so is Y. The Attempt at a Solution I used the following lemma: X is normal iff given a closed set A and open set U containing A, there is an open set V containing A and whose...
  22. H

    An application of the closed graph theorem.

    So I have to show that a projection P (i.e a linear operator with P=P²) on a Banach space X is bounded if and only if \ker (P) and P(X) are closed subspaces of X.My idea was to boil it down, using the closed graph theorem. What's left for me now is to show that the graph G(P):=\{(x,y)\in X\times...
  23. A

    Exploring Pressure from Speaker Motion in Open & Closed Spaces

    I've been studying the behaviour of speakers and headphones for my own interest. I was able to derive the differential equation governing the motion of the speaker itself, what I've had trouble doing is deriving the pressure created by the speaker. First the equation I've derived for...
  24. radou

    Showing a closed subspace of a Lindelöf space is Lindelöf

    Homework Statement As the title says, one needs to show that if A is a closed subspace of a Lindelöf space X, then A is itself Lindelöf. The Attempt at a Solution Let U be an open covering for the subspace A. (An open covering for a set S is a collection of open sets whose union equals...
  25. E

    Two separate barycenters of a binary system in a small closed universe?

    EDITED: I decided to move the thought experiment that led me to this question to the bottom of this thread. I will first state what my question is. Suppose that we live in a closed universe of three spatial dimensions and this universe is in a state of rapid collapse. Now suppose that all of...
  26. A

    Calculating the magnetic field inside a closed loop of wire

    Homework Statement I basically have a closed loop of wire with a current flowing through it clockwise. I need to determine the magnetic field in the center, along the z-axis. Homework Equations Ampere's Law and Biot-Savart Law The Attempt at a Solution Using Biot-Savart Law I...
  27. G

    Defining Closed, Open, and Compact Sets in R^n

    Homework Statement How to define closed,, open and compact sets?Are they bounded or not? Homework Equations For example {x,y:1<x<2} The Attempt at a Solution It's is opened as all points are inner Can you please say the rule for defining the type of the set? Like for example...
  28. K

    Is Every Convergent Sequence in a Closed Set a Cauchy Sequence?

    Homework Statement A set F\subseteqR is closed iff every Cauchy sequence contained in F has a limit that is also an element of F. Homework Equations The Attempt at a Solution Let F be closed. Then F contains its limit points. This means x=lima_{n} are elements of F.
  29. B

    Statistical behaviour of ideal particles in a closed box

    Suppose I have N ideal particles in an enclosure, be it a ball or a cube or some other form. The particles shall bounce off the walls of the enclosure and against each other without losing speed. The velocity of each particle i shall be such that it fullfills |v_i|=\rho, where \rho is constant...
  30. G

    Finite field is algebraically closed under constraint?

    A field K is called algebraically closed field if any no-zero polynomial has at least one root in K. Given finite field F_q, q=p^m, p is a prime and m is non-negative integer. A famous property of finite field is any element in F_q satisfies: x^q=x. Then I have such an assumption...
  31. M

    Example of a linear subset of Hilbert space that is not closed

    Homework Statement Prove that for a linear set M a subset of Hilbert space, that the set perpendicular to the set perpendicular to M is equal to M iff M is closed. The Attempt at a Solution I already have my proof -- but what is an example of a linear subset of H that is not closed? I think...
  32. J

    Prove that any finite set is closed

    Homework Statement As the title says Homework Equations Definitions of "open" and "closed" The Attempt at a Solution Suppose a finite set S is not closed. Then Sc is not open, and there exists an element x of Sc, so that for all µ > 0, either x + u, or x - u, is an element of S...
  33. R

    Prove that SU(n) is closed and bounded

    Homework Statement Prove that SU(n) is closed and bounded Homework Equations The Attempt at a Solution So in order to prove this, I first mapped SU(n) to be a subset of R^{{2n}^2}. To prove the closed portion, I tried mapping a sequence in SU(n) to a sequence in R^{{2n}^2}. However, I...
  34. R

    Open and closed intervals and real numbers

    Homework Statement Show that: Let S be a subset of the real numbers such that S is bounded above and below and if some x and y are in S with x not equal to y, then all numbers between x and y are in S. then there exist unique numbers a and b in R with a<b such that S is one of the...
  35. M

    Buzzer for a normally closed switch

    I'm sure there is an easy solution to my problem, but I very little electronics background and don't yet have a breadboard to even experiment on. In fact, it was only two days ago that I learned a 555 doesn't need to (or can) be programmed by a computer to use. I want to make a buzzer that...
  36. R

    The set of limit points is closed

    Homework Statement L is the set of limit point of A in the real space, prove that L is closed. Homework Equations The Attempt at a Solution L may or may not have limit points. If L does not have limit points, then it's obviously closed. If L has limit points, the let l be a...
  37. C

    Subspace of l2/L2 that is closed/not closed.

    Homework Statement Give a nontrivial example of an infinite dimensional subspace in l2(R) that is closed. Also find an example of an infinite dimensional subspace of l2(R) that is not closed. Repeat the same two questions for L2(R). The Attempt at a Solution To my understanding, l2 is...
  38. D

    Exploring the Relationship Between Open and Closed Sets in Topology

    Homework Statement Prove that if S is open and Sc is open then boundary of S must be empty The Attempt at a Solution S is open means boundary of S is a subset of Sc Sc is open means boundary of Sc is a subset of S (By taking complement of both sides from the definition ?) This means that they...
  39. radou

    Showing a set is closed with the definition of continuity

    Homework Statement I need to show that the subset of R^2 given with A = {(x, y) : xy = 1} is closed by using the "closed set formulation" of continuity. The Attempt at a Solution So, if a function f : X --> Y is continuous, then for every closed subset B of Y, its preimage f^-1(B) is...
  40. K

    Is the rate of entropy of a closed system constant?

    If I am within that closed system, can I make the rate of entropy of the entire system any faster or slower overall? Does the existence of life within the system decrease or increase the overall system entropy rate?
  41. M

    Discover the Derivation of Closed Form Summation | Step-by-Step Explanation

    How were they able to derive this?
  42. B

    Closed and Open sets in R (or 'clopen')

    I'm sure this has been asked before, but the proofs I've seen use the fact R is connected or continuous functions is some way. I'm trying to prove it with the things that have only been presented in the book so far (Mathematical Analysis by Apostol). So, let A be a subset of R which is both...
  43. B

    Closed, bounded but not compact

    Homework Statement let |e-x-e-y| be a metric, x,y over R. let X=[0,infinity) be a metric space. prove that X is closed, bounded but not compact. Homework Equations The Attempt at a Solution there is no problem for me to show that X is closed and bounded. but how do I prove...
  44. S

    How to Solve Dot Product Homework Using Vector v?

    Homework Statement Homework Equations know dot product The Attempt at a Solution PART A PART B not sure what's it asking for help would be great
  45. B

    Cl(A) smallest closed set containing A.

    My professor proved this result in class, but I don't understand the "simple" direction. He said that the above result is in another words proving that Cl(a) = intersection of all closed sets that contain A. So he proved Cl(A) subset of intersection of all of the closed sets containing A...
  46. A

    Is the universe a closed system?

    Would the conservation of mass apply to the universe? Sorry I looked for awhile for the answer to this question but I couldn't really find anything.
  47. M

    Closed System and Center of Mass

    Hello, Can a closed system change the position of its center of mass if no external force is exerted on it?
  48. S

    Is the interval I of an autonomous diff closed?

    Is the interval I of an autonomous diff closed? Homework Statement Given this autonomous diff.eqn Where we have an open set E defined on R^n and f \in \mathcal{C}^1(E) x' = f(x) where x(t_a) = x(t_b) and t_a,t_b \in I and where t_a < t_b. Show for n = 1, that the solution x is...
  49. M

    How to Convert Finite Sums to Closed Form with Limit |a| < 1

    Ignore the above, I was haveing problems with the symbol... Convert each to closed form: 1. Sum from i=1 to n of: \frac{n}{a^n} 2. Sum from i=1 to n of: \frac{1}{a^n} Thanks. P.S. I know how to do it if it was an infinite series, but not for this.
  50. P

    Is this considered a closed expression for pi?

    ln(-1)/i=pi this equation does not use limits or integrals, as you can see, but it does involve imaginary numbers. Does this make it an open expression, or does the fact that it uses i not matter?
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