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. L

    Proving Space of Differential Functions Not Closed

    how i prove that space of defrential function not closed?
  2. Somefantastik

    Complete, Equivalent, Closed sets

    If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete? What if it is known that A is closed, can it then be said that B is also closed?
  3. S

    Understanding Harmonics and Overtones in Closed and Open Pipe Organ Designs

    This isn't actually a homework/coursework question, but rather a need to clarify a discepancy between my lecturer's notes and a textbook. My lecturer's notes state that for an "organ" pipe, closed at one end, the 1st harmonic frequency will be 4L. For the 2nd harmonic the frequency will be...
  4. A

    Boiling point of water in a closed container

    Hi all, for a closed container with water under pressure in it, let's say GAUGE PRESSURE of the container is 101 kPa, what will the boiling point of water in that vessel be? Will it be 100 deg C, or will it be 120 deg C because the ABSOLUTE PRESSURE of the water in that vessel is 202kPa...
  5. J

    Find closed formula for the sequence.

    I have a problem with this question. The question is "Find closed formula for the sequence 0,1,3,0,1,3,0,1,3..." It can be written as (0,1,0,0,1,0,0,1,0...)+3*(0,0,1,0,0,1,0,0,1...) I know the sequence of 0,0,1,0,0,1..., but I do not know how to get the sequence of 0,1,0,0,1,0... And is...
  6. B

    Proving Closed Sets using the Sequential Criterion

    I'm sorry if this should be in the Analysis forum; I figured it pertained to topology though. Let Y be a subspace of a metric space (X,d) and let A be a subset of Y. The proposition includes conditions for A to be open or closed in Y. In class the teacher first proved when A is open and then...
  7. I

    Riemann surface of four closed string interaction (Zwiebach section26.5)

    There are three kinds of light-cone string diagrams for four closed string interactions. As displayed by fig. 26.10, 26.11 and 26.12 of section 26.5 of Zwiebach's book. For each light-cone string diagram, it is characterized by two parameters, the time difference of the two interaction...
  8. K

    Closure & Closed Sets in metric space

    Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F. Claim 1: F is contained in the clousre of F...
  9. T

    Exploring Closed Spaces: Negative Curvature and the Inside of a Torus

    In my cosmology lectures they say that a negative curvature gives an infinite space but I was thinking what about the inside of a torus. Isn't that a closed space too? Cant any value of k apart from 0 result in a closed space??
  10. K

    Proving Closure and Openness in Metric Spaces

    "Closed" set in a metric space Homework Statement 1) Let (X,d) be a metric space. Prove that a "closed" ball {x E X: d(x,a) ≤ r} is a closed set. [SOLVED] 2) Suppose that (xn) is a sequence in a metric space X such that lim xn = a exists. Prove that {xn: n E N} U {a} is a closed subset of...
  11. J

    Energy conservation in a closed universe

    Consider a 4 dimensional spacetime which is everywhere flat and is closed in along all three spatial dimensions. Since spacetime is everywhere flat we can use global inertial frames. We will not consider any gravitational interactions in this problem. Now a valid vacuum solution to maxwell's...
  12. M

    Is [0, infinity) a closed set?

    Homework Statement Is [0, infinity) a closed set? Homework Equations N/A The Attempt at a Solution It's easy to say that its not. But the solution in my textbook suggests otherwise. Why is this so? Thanks! M
  13. B

    Mode Expansion of Closed String with Compact Dimensions

    Im working through derivations of string equations of motion from the Nambu-Goto Action and I'm stuck on something that I think must be trivial, just a math step that I can't really see how to work through. At this point I've derived the equation of motion for the closed string from the wave...
  14. K

    Decreasing sequence of closed balls in COMPLETE metric space

    Homework Statement Give an example of a decreasing sequence of closed balls in a complete metric space with empty intersection. Hint 1: use a metric on N topologically equivalent to the discrete metric so that {n≥k} are closed balls. In={n,n+1,n+2,...}. Homework Equations N/A The...
  15. D

    Twin paradox in a closed universe

    I`ve thought about a special sort of twin paradox. I know the usual explanation of the twin paradox but give me please the answer to this special case: Imagine: A static universe (non-expanding) with a closed geometry and a circumference of one lightyear. The twins start their journey in...
  16. J

    Find the charge on the plates a long time after the switch is closed

    Homework Statement I'm having a problem with the circuit in the attached diagram. I am looking for the charge on the plates of the capacitor a long time after the switch is closed. Homework Equations The Attempt at a Solution I found the current leaving the battery is 0.962 A a...
  17. M

    Whats the difference between a closed set vs a set thats NOT open?

    Let f: D->R be continuous. If D is not open, then f(D) is not open. Why can they not replace 'not open' with closed? Thank you M
  18. B

    No Quadrilateral/Pentagon Knots: Simple Closed Polygons are Trivial

    Hi, everyone: I would appreciate any help with the following: I am trying to refresh my knot theory--it's been a while. I am trying to answer the following: 1) Every simple polygonal knot P in R^2 is trivial.: I have tried to actually construct a...
  19. F

    Proving that (6k+1) is Closed Under Multiplication

    Show the progression (6k +1) (k is an integer) is closed under multiplication: Firstly I should check that I remember what this means... If it is closed when you multiply any 2 elements together you get an element that is in the set? So for this I thought just show (6k+1)(6n+1), where k...
  20. E

    Is it possible to make a closed mirror system?

    Hello, my name is Edward Solomon, after much experimentation and calculation I have failed to make a system that can reflect light in a closed system. Now I am not naive enough to believe I can make a true closed system. There is an absorption and conversion to heat each time light strikes a...
  21. S

    Understanding Open and Closed Sets in Topology

    I'm self studying topology and so I don't have much direction, however I found this wonderful little pdf called topology without tears. So to get to the meat of the question, given that \tau is a topology on the set X giving (\tau,X), the members of \tau are called open sets. Up to that point...
  22. S

    A basic question: What does closed form mean?

    A basic question: What does "closed form" mean? "The point here is that \sigma algebras are difficult but \pi systems are easy: one can often write down in closed form the general element of a \pi system while the general element event of \mathbf B \mathbf is impossibly complicated" - From the...
  23. S

    Convergence of Sequences and closed sets

    Homework Statement This is the Theorem as stated in the book: Let S be a subset of a metric space E. Then S is closed if and only if, whenever p1, p2, p3,... is a sequence of points of S that is convergent in E, we have: lim(n->inf)pn is in S. Homework Equations From "introduction to...
  24. N

    Differentiability on a closed interval

    Homework Statement Hi all I wish to show differentiability of g(x)=x on the interval [-pi, pi]. This is what I have done: g'(a) = \mathop {\lim }\limits_{h \to 0} \frac{{g\left( {a + h} \right) - g\left( {a} \right)}}{h} \\ = \mathop {\lim }\limits_{h \to 0} \frac{h}{h} \\ = 1...
  25. D

    Prove that countable intersections of closed subset of R^d are closed

    Prove that countable intersections of closed subset of R^d are closed
  26. S

    Closed set equivalence theorem

    Homework Statement Hi guys, this problem gave me some trouble before, but I'd like to know if I have it worked out now... "If S = S\cupBdyS, then S is closed (S_{compliment} is open) Homework Equations S is equal to it's closure. The Attempt at a Solution 1. Pick a point p in...
  27. J

    Connectedness of the closed interval

    Hi, I'm studying for the final exam in my first course in topology. I'm currently recalling as many theorems as I can and trying to prove them without referring to a text or notes. I think I have a proof that the closed interval [0,1] is connected, but it's different than what I have in my...
  28. R

    Closed system piston cylinder device problem

    Homework Statement A closed system comprising a cylinder and frictionless piston contains 1kg of a perfect gas of which molecular mass is 26. The piston is loaded so that the pressure is constant at 200kPa. Heat is supplied causing the gas to expand from 0.5m^3 to 1m^3. Calculate heat...
  29. J

    Proving Closed Rectangle A is a Closed Set

    Homework Statement Prove that a closed rectangle A \subset \mathbb{R}^n is a closed set. Homework Equations N/A The Attempt at a Solution Let A = [a_1,b_1] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n, then A is closed if and only if its complement, \mathbb{R}^n - A, is...
  30. B

    Calculating Gauge Pressure in a Closed Tube

    Homework Statement The container shown in the figure is filled with oil. It is open to the atmosphere on the left. What is the gauge pressure at point A? Point A is 50cm high from the ground and 50 cm from the top. Homework Equations po+(density)gd The Attempt at a Solution I...
  31. S

    Homomorphisms into an Algebraically Closed Field

    Okay, so I'm trying to finish of a problem on integral closure and I am rather unsure if the following fact is true: If L embeds into an algebraically closed field K and F is an algebraic extension of L, then it is possible to extend the embedding of L to F into K. Now the case where F...
  32. R

    Proving Gauss's Theorem: Closed Surface

    Homework Statement Using Gauss's theorem prove that \int_{s}\vec{n}ds=0 if s is a closed surface. Homework Equations Gauss's theorem: \int_{V}\nabla.\vec{A}dv= \oint_{s}\vec{A}.\vec{n}da The Attempt at a Solution In this problem \vec{A} is constant so \nabla.\vec{A}=0 so...
  33. A

    Metric Space, open and closed sets

    Homework Statement Let X be set donoted by the discrete metrics d(x; y) =(0 if x = y; 1 if x not equal y: (a) Show that any sub set Y of X is open in X (b) Show that any sub set Y of X is closed in y Homework Equations In a topological space, a set is closed if and only if it...
  34. A

    Metric Space, closed ball is a closed set. prove this

    Homework Statement Let (X, d) be a metric space. The set Y in X , d(x; y) less than equal to r is called a closed set with radius r centred at point X. Show that a closed ball is a closed set. Homework Equations In a topological space, a set is closed if and only if it coincides...
  35. P

    Proving the Intersection of Closed Sets is Closed | Homework Solution

    Homework Statement Show that the intersection of two closed sets is closed. Homework Equations The Attempt at a Solution Let X and Y be closed sets i.e. X and Y are equal to their closure X_ and Y_. Then X\capY is equal to X_\capY_.
  36. U

    Sum of a closed set and a compact set, closed?

    Homework Statement I am trying to prove that, if X is compact and Y is closed, X+Y is closed. Both X and Y are sets of real numbers. Homework Equations The Attempt at a Solution I know that a sum of two closed sets isn't necessarily closed. So I presume the key must be the...
  37. K

    Does every continuous function has a power series expansion on a closed interval

    By Weierstrass approximation theorem, it seems to be obvious that every continuous function has a power expansion on a closed interval, but I'm not 100% sure about this. Is this genuinely true or there're some counterexamples?
  38. S

    Voltage in capacitor after switch has been closed

    The drawing shows two fully charged capacitors (C1 = 2.00\muF, q1 = 6.00\muC; C2 = 8.00\muF, q=12.0\muC). The switch is closed, and charge flows until equilibrium is reestablished (i.e., until both capacitors have the sam voltage across their plates). Find the resulting voltage across either...
  39. F

    Show a closed subset of a compact set is also compact

    Homework Statement Show that if E is a closed subset of a compact set F, then E is also compact. Homework Equations I'm pretty sure you refer back to the Heine-Borel theorem to do this. "A subset of E of Rk is compact iff it is closed and bounded" The Attempt at a Solution We...
  40. W

    Proving Closure of Set T: f(x)=g(x) on Closed Domain [a,b] in R

    Suppose f:[a,b]--> R and g:[a,b]-->R. Let T={x:f(x)=g(x)} Prove that T is closed. I know that a closed set is one which contains all of its accumulation points. I know that f and g must be uniformly continuous since they have compact domains, that is, closed and bounded domains. Now T is the...
  41. D

    Solution: Open & Closed Subsets in R: Na(E) Nonempty, [a-x,a+x] ⊆ E, E = R

    Homework Statement Let E be a nonempty subset of R, and assume that E is both open and closed. Since E is nonempty there is an element a \in E. De note the set Na(E) = {x > 0|(a-x, a+x) \subsetE} (a) Explain why Na(E) is nonempty. (b) Prove that if x \in Na(E) then [a-x, a+x] \subset...
  42. Saladsamurai

    Gauss's TheoremNet Force due to Uniform Pressure on a Closed Surface = 0

    Homework Statement In my fluid mechanics text, it states that the Net Force due to a uniform pressure acting on a closed surface is zero or: \mathbf{F} = \int_{surface}p(\mathbf{-n})\,dA = 0 \,\,\,\,\,\,\,(1) where n is the unit normal vector and is defined as positive pointing outward from...
  43. T

    Closed set, compact set, and a definition of distance between sets

    Homework Statement Let E and F be 2 non-empty subsets of R^{n}. Define the distance between E and F as follows: d(E,F) = inf_{x\in E , y\in F} | x - y | (a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E...
  44. R

    Calculate Flux in a closed Triangle

    Homework Statement Consider a closed triangular box resting within a horizontal electric field of magnitude E = 7.80 & 104 N/C as shown in Figure P24.4. Calculate the electric flux through (a) the vertical rectangular surface, (b) the slanted surface, and (c) the entire surface of the box...
  45. R

    Prove if S is Open and Closed it must be Rn

    The main question: Let S be a subset in Rn which is both open and closed. If S is non-empty, prove that S= Rn. I am allowed to assume Rn is convex. Things I've considered and worked with: The compliment of Rn is an empty set which has no boundaries and therefore neither does Rn. Therefore...
  46. K

    Integral of a closed surface over a general region

    I have been working on this problem for a few hours and am completely stuck. It seems like a simple problem to me but when I attempt it I get nowhere. The problem is: Show that \frac{1}{3}\oint\oint_{S}\vec{r} \cdot d\vec{s} = V where V is the volume enclosed by the closed surface S=...
  47. S

    Constructing a Bounded Closed set

    Homework Statement i) Construct a bounded closed subset of R (reals) with exactly three limit points ii) Construct a bounded closed set E contained in R for which E' (set of limit points of E) is a countable set. Homework Equations Definition of limit point used: Let A be a subset of...
  48. A

    Is \{1,2,3,4,5\ldots\} a Closed Set in \mathbb{R}?

    The set \{1,2,3,4,5\ldots\}...is it closed as a subset of \mathbb{R}? I'm thinking "yes," but I'm unsure of myself for some reason. (And yes, this is just the set of positive integers.
  49. S

    If a set A is both open and closed then it is R(set of real numbers)

    if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way
  50. R

    Show that a hyperplane in R^n is a closed set

    Homework Statement Show that a hyperplane in Rn is a closed set Homework Equations The Attempt at a Solution I was thinking maybe try to prove that the complement of the hyperplane is open?
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