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. M

    Efficient Steps for Closed Form Summation: A Detailed Guide | 1.7^k and 2k"

    So here are my steps, which for some reason I feel are very wrong: Well in closed form would be [n(n+1)]/2 so 2k would be 2*[n(n+1)]/2 For 1.7^k, I used a different form, which I don't have the formula for in front of me, but the final result for that part is [1.7^(n+1) - 1] /[1.7 - 1] So...
  2. I

    Cancellative set in a semiring that is not multiplicatively closed

    Definition: A semigroup is a pair (R,op) where R is a set an op is a binary operation that is closed and associative. A commutative semigroup is a semigroup where op satisfies for all a,b in R, op(a,b) = op(b,a). A monoid is a semigroup where with an identity,e, for op, satisfying for all r in...
  3. M

    Analysis: the closure of a set is closed?

    Homework Statement Prove or disprove the following statement: The closure of a set S is closed. Homework Equations Definition of closure: set T is the closure of set S means that T is the union of S and the set of limit points of S. Definition of a closed set: set S is...
  4. M

    Set of all the limit points of Set E. Prove that its closed.

    Is it correct to make the following statement? If a point x in E is not a limit point of E, then any neighborhood V of x will--at most--contain finitely many points of E. Thus, its possible for V to contain only one point, namely, x. Thanks, M
  5. P

    Closed Wire Loop: Induced Current & Dissipated Energy

    Homework Statement 5 A closed wire loop in the form of a square of side 4.0 cm is mounted with its plane horizontal. The loop has a resistance of 2.0 x 10-3Ω, and negligible self-inductance. The loop is situated in a magnetic field of 0.70 T, directed vertically downwards. When the field is...
  6. E

    Velocity of sound in air in a closed air column

    Am I missing something and doing this all wrong? Experiment and data: The level of water in the glass tube was adjusted by raising the supply tank until the sound from the tuning fork was at its loudest. This level corresponds to first resonance position and it was recorded. The...
  7. G

    Find a closed interval topology

    Homework Statement Let X be an ordered set where every closed interval is compact. Prove that X has the least upper bound property. Homework Equations X having the least upper bound property means that every nonempty subset that is bounded from above has a least upper bound, in other...
  8. T

    Free damped pendulum solution : closed form

    I am modelling a free damped simple pendulum and was wondering if anyone could refer me to a paper or perhaps provide me with an expression describing the motion of the pendlum with large initial amplitude. I have solved the equations numerically but am implementing an optimization routine and...
  9. T

    Closed form equations approximating the motion of a free undamped pendulum

    I am modelling a free damped simple pendulum and was wondering if anyone could refer me to a paper or perhaps provide me with an expression describing the motion of the pendlum with large initial amplitude. I have solved the equations numerically but am implementing an optimization routine and...
  10. Z

    Has the Classical Motion of Closed String Loops Been Explored in String Theory?

    Has anyone ever seen the treatment of a closed classical string loop. Like if you had a loop of string on the space shuttle and subject it to accoustic driving or initial impulses. I post this here in beyond the standard model because no one in the classical physics section seems to have heard...
  11. Z

    Closed loop of classical string

    Has anyone seen a treatment of how to use the wave equation to describe a closed loop of string. I am talking ordinary strings here not the fancy string theory kind.
  12. J

    Any closed interval [a,b] is compact ?

    Hi All, So all closed interval [a,b] is compact (see Theorem 2.2.1 in Real Analysis and Probability by RM Dudley) Now, Let's say I have [0,10] as my closed interval. Let My Open Cover be (0, 5) (5, 7.5) (7.5, 8.75) (8.75, 9.375) ... Essentially, The length of each open...
  13. A

    Proof involving a closed set of integers

    Homework Statement proove is either true of false let A be a set of integer closed under subtraction. if x and y are element of A, then x-ny is in A for any n in Z. Homework Equations n/a The Attempt at a Solution is there any proof, without induction? i suspect its true because any...
  14. K

    Heat loss due to Evaporation in a closed environment

    I am working on an incubator/shaker for laboratory use. I am trying to work out a temperature failure and repair it, but that is besides the point here. I was looking through the user manual to try to get some clues about the failure and I came across this: "Depending on various conditions...
  15. E

    Find the maximum and minimum dimension of a closed loop

    Dear all, Is there a method to find the maximum and minimum dimension of an irregular closed loop? This is a problem when we want to define the full-width - half maximum of a image. The level contour of this image at its half maximum can be an irregular closed loop. Any reference or...
  16. A

    Liquid in a vertical pipe open at the lower end and closed at the top

    Consider a vertical pipe partially filled with liquid. The pipe is open at the lower end and closed at the top. See the attached picture. Will the liquid fall out or not? In a small diameter pipe a stable meniscus will form due to surface tension and prevent the water from falling out. In a...
  17. radou

    Is the Graph of a Continuous Function Closed if the Spaces are T2 and T1?

    Me again. Problem. Let X be a topological space, and Y a T2-space (i.e. a Haussdorf topological space). Let f : X --> Y be a continuous function. One needs to show that the graph of , i.e. the set G = {(x, f(x)) : x is in X} is closed in X x Y. Attempt of proof. To show what we need to show...
  18. W

    Open and closed in the geometrical sense vs the thermodynamic sense

    "Open" and "closed" in the geometrical sense vs the thermodynamic sense Perhaps this is a silly question, but what is the relationship between the words "open" and "closed" in the geometrical sense (open, flat, closed universes) and in the thermodynamic sense (open and closed systems) in the...
  19. B

    Elephant Toothpaste Experiment in closed container used to compress air

    In case you have never heard of the elephant toothpaste experiment, take a look at this: http://www.using-hydrogen-peroxide.com/elephant-toothpaste.html" I was just wondering, if you put the chemicals together in an air tight container, would the air pressure increase?
  20. V

    Determining Flow Rate from a Closed Pipe

    im trying top find the flow rate of water from a closed pipe. one thing that i think i can work with is a hose. there is a hose branching of the main flow pipe, which can be used to clean floor etc. im thinking that i can not open the hose and thus measure the stagnatiob pressure at the...
  21. radou

    Why is r/2 used in the proof for one point set being closed?

    So, I'm going through a proposition, which states that if (X, d) is a metric space, then any set {x}, where x e X, is a closed subset of X. First of all, could we do this proof to assume the contrary? Since then obviously for the point x from {x} there doesn't exist any real number r > 0 such...
  22. M

    Compact sets in Hausdorff space are closed

    First of all I just want to rant why is the Latex preview feature such a complete failure in Firefox? Actually it is really bad and buggy in IE too... So I am reading into Foundations of geometry by Abraham and Marsden and there is a basic topology proof that's giving me some trouble. They...
  23. H

    Proving: Closed Curve Integral in 3D Space

    Homework Statement Giving 2 closed curves in 3-dimension space C1 and C2, prove that:\oint _{C1} \oint _{C2}\frac{(\vec{dl_2}.\hat{r_{12}})\vec{dl_1}}{r^2_{12}}=0 Where: _ \vec{dl_1} and \vec{dl_2} are the vector elements of the curves C1 and C2 respectively. _ r_{12} is the distance between...
  24. S

    Analyzing LED Circuit Connections with Switch S Closed

    Homework Statement An LED is connected as shown (see attached) When switch S is closed: A. the p-n junction is reverse biased and free charge carriers are produced which may recombine to give quanta of radiation. B. the p-n junction is forward biased and positive and negative charge...
  25. S

    Why does work done by a conservative force = 0 in a closed path?

    Why does work done by a conservative force = 0 in a closed path? I know this sounds foolish :rolleyes: but how can some forces have such a property? Can anybody give a satisfactory physical explanation?:confused:
  26. Q

    Ampere's law for a closed ring bar magnet

    Homework Statement A long bar magnet is bent into the form of a closed ring. If the intensity of magnetisation is M, and ignoring any end effects due to the join, find the magnetic field H and the induction B: (a) Inside the material of the magnet (b) just outside Homework Equations...
  27. D

    Solved: Closed Form Solution for SIGMA e^(i/n)

    Homework Statement Find the closed form value for n SIGMA e^(i/n) i= 0 Homework Equations ? The Attempt at a Solution summation expands to 1 + e^(1/n) + e^(2/n) - - - - - e^1 To be honest i have no clue how to go about these kinds of problems so a general help would...
  28. S

    Using Closed/Open Balls in Rosenlicht's Intro to Analysis Proofs

    Homework Statement In Rosenlicht's Intro to Analysis, there is a proposition (p. 52). A Cauchy sequence of points in a metric space is bounded. Proof: For if the sequence is P1, P2, P3, ... and ε is any positive number and N an integer such tat d(Pn, Pm) < ε if n, m > N, then for any...
  29. S

    Open and closed sets of metric space

    Homework Statement I am using Rosenlicht's Intro to Analysis to self-study. 1.) I learn that the complements of an open ball is a closed ball. And... 2.) Some subsets of metric space are neither open nor closed. Homework Equations Is something amiss here? I do not understand how...
  30. M

    If the universe is closed - scenario question

    First post, please excuse my ignorance. If the Universe is closed, then at the end of the expansion, micro gravity eventually pulls all objects together. Black holes absorb more and more stars and whole galaxies and eventually each other until there is just one black hole and no matter left...
  31. Somefantastik

    Bounded & Closed Set: A = \{(x,y): 0\leq xy \leq 1\}

    Homework Statement A = \left\{(x,y): 0\leq xy \leq 1\right\}, A \in R^{2} I'm trying to determine if this set is bounded and/or closed. Homework Equations if X = (x,y) euclidean metric: ||X|| = \sqrt{x^{2}+y^{2}} The Attempt at a Solution I know a bounded set =>...
  32. Somefantastik

    Analyzing a Closed Set on the Complex Line

    Homework Statement on the complex line, with the usual metric, I need to determine if this is a closed set. A = \left\{\left|\frac{1}{z^{2}+1} \right|: |z| = 1 ; z\neq \pm i\right \} Homework Equations The Attempt at a Solution A closed set implies that the set of all limit points belongs...
  33. C

    Product topology, closed subset, Hausdorff

    Homework Statement Let (X,\tau_X) and (Y,\tau_Y) be topological spaces, and let f : X \to Y be continuous. Let Y be Hausdorff, and prove that the graph of f i.e. \graph(f) := \{ (x,f(x)) | x \in X \} is a closed subset of X \times Y. Homework Equations The Attempt at a Solution...
  34. C

    Open sets and closed sets in product topology

    Homework Statement Let (X_a, \tau_a), a \in A be topological spaces, and let \displaystyle X = \prod_{a \in A} X_a. Homework Equations 1. Prove that the projection maps p_a : X \to X_a are open maps. 2. Let S_a \subseteq X_a and let \displaystyle S = \prod_{a \in A} S_a \subseteq...
  35. R

    Compactness of closed unit ball

    Homework Statement Let l∞ be the space of bounded sequences of real numbers, endowed with the norm ∥x∥∞ = supn∈N |xn | , where x = (xn )n∈N . Prove that the closed unit ball of l∞ , B(0, 1) = {x ∈ l∞ ; ∥x∥∞ ≤ 1} , is not compact. Homework Equations The Attempt at a Solution I'm...
  36. T

    Closed curve line integral of gradient using Green's Theorem

    Apostol page 386, problem 5 Homework Statement Given f,g continuously differentiable on open connected S in the plane, show \oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha} for any piecewise Jordan curve C. Homework Equations 1. Green's Theorem 2. \frac{\partial...
  37. C

    Singleton sets closed in T_1 and Hausdorff spaces

    If (X,\tau) is either a T_1 space or Hausdorff space then for any x \in X the singleton set \{ x \} is closed. Why is this the case? I can't see the reason from the definitions of the spaces. Definition: Let (X,\tau) be a topological space and let x,y \in X be any two distinct points, if...
  38. T

    Projectile Launcher, large current, closed loop, magnetic field

    Homework Statement A projecticle launcher is shown in the attachment. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar touching the rails. A magnetic field is perpendicular to the plane of the circuit. If the bar has a...
  39. J

    Organ Pipe Harmonic Frequencies and Pipe Length Calculation

    Homework Statement An organ pipe has two successive harmonics with frequencies 1760 Hz and 2160 Hz. Is this an open or stopped pipe? Which two harmonics are these? What is the length of the pipe? Homework Equations L=v/2f The Attempt at a Solution I'm really just having trouble...
  40. K

    Closed subset of R^n has an element of minimal norm

    Homework Statement a) Let V be a normed vector space. Then show that (by the triangle inequality) the function f(x)=||x|| is a Lipschitz function from V into [0,∞). In particular, f is uniformly continuous on V. b) Show that a closed subset F of contains an element of minimal norm, that...
  41. H_man

    Putting a discrete sum of cosines in closed form

    Homework Statement I've just found what I think is the Green's function for a source between two ideal conducting planes at x = 0 and x = l:Homework Equations G(x,x') = \Sigma \frac{icos(\pi n x/l)}{(\pi n /l)} The Attempt at a Solution The question then wants me to put...
  42. H

    Question about Flux through a closed surface

    I understand that magnetic flux through a closed surface is zero, but what is the exact definition of a closed surface? The textbook I'm using is rather vague with this definition and I want to make sure I have the definition nailed down for the exam in case my professor tries anything tricky.
  43. H

    Standing Waves in a closed pipe

    So confused about standing waves in a closed pipe, which is open at one end and closed at the other. The closed end has a node while the open end has an antinode. To figure the wavelength, i use the formula: Lambda = 4L/n where n is the number of harmonic and can only be odd integers...
  44. T

    Commutative and Associative Addition in Closed Sets: A Conceptual Explanation

    I'm reading Riley's "Mathematical Methods for Physics and Engineering" and I came across this expression about vector spaces: "A set of objects (vectors) a, b, c, ... is said to form a linear vector space V if the set is closed under commutative and associative addition (...)" What I don't...
  45. A

    An open mapping is not necessarily a closed mapping in functional analysis

    We know that a linear operator T:X\rightarrowY between two Banach Spaces X and Y is an open mapping if T is surjective. Here open mapping means that T sends open subsets of X to open subsets of Y. Prove that if T is an open mapping between two Banach Spaces then it is not necessarily a closed...
  46. M

    How Does Pipe Closure Affect Organ Pitch and Length?

    [b]1. the lowest note on an organ is 16.4 Hz. What is the shortest open organ pipe that will resonate at this frequnecy? What would be the pitch if the same organ pipe were closed? [b]3. is the answer 32.8 meters and 65.6 meters?
  47. E

    Cos(x) is not closed function?

    I think cos(x) is closed function in R. But I heard that cos(x) is not closed function in R. What do I choose closed set A in R, cos(A) is not closed in R? Help...
  48. M

    Calculating Liquid Density in a Closed Circuit

    how can i calculate density of a liquid pumped in a closed circuit and i don't know the nature of it ?
  49. M

    Closed Orbit of a Flow on a Manifold

    Homework Statement Let γ be a closed orbit of the flow φ on the manifold M and suppose there exists T>0 and X0 є γ such that φT(X0) = X0. Prove that φT(X) = X for every X0 є γ. Furthermore locate two closed orbits γ1 and γ2 and positive periods T1 and T2 for the flow of r ̇=r(r-1)(r-2); θ...
  50. P

    Closed form expression for f(x) = sigma (n = 1 to infinity) for x^n / [n(n+1)]

    Homework Statement Consider the power series. sigma (n=1 to infinity) x^n / [n(n+1)] if f(x) = sigma x^n / [n(n+1)], then compute a closed-form expression for f(x). It says: "Hint: let g(x) = x * f(x) and compute g''(x). Integrate this twice to get back to g(x) and hence derive...
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