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. Math Amateur

    MHB Outer measure of a closed interval .... Axler, Result 2.14 ....

    I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help with the proof of Result 2.14 ... Result 2.14 and its proof read as follows: In the above proof by Axler we read the following: " ... ... We will now prove by...
  2. LCSphysicist

    Solving a Closed System w/Conservative Forces: Is E1=E2 Always True?

    This problem is very easy to solve considering that the two particles belong a closed system under action of conservatives force. My doubt is if it is possible to solve the problem by consider one particle by time, that is: Suppose that we know the particle m one is under gravitational force...
  3. U

    I Big bounce theory in light of the closed Universe finding

    As you all know: Planck evidence for a closed Universe and a possible crisis for cosmology https://www.nature.com/articles/s41550-019-0906-9 If confirmed, it would bring back the Big Bounce as a possible hypothesis for the evolution of the universe. This has long-term consequences being that...
  4. Frigus

    Why is the net flux through a closed surface equal to zero?

    Suppose we have placed a cube in field which varies linearly with z axis so electric field magnitude on coordinates of face ABCD is clearly more than face EFGH and we know area of both faces are equal, So if we calculate flux then it would be non zero but it contradicts with the fact that...
  5. U

    I If the Universe is closed, is it thereby self-contained as well?

    https://www.google.com/amp/s/phys.org/news/2019-11-universe-rethink-cosmos.ampWhat do the results of the closed universe study tell us in terms of past cosmic sequences, if it is indeed the proper description of the universe? Would it entail a self-contained universe? By self-contained I mean...
  6. C

    Circuits and the percent change of current when the switch is closed

    For an ideal battery (r = 0 Ω), closing the switch in (Figure 1)does not affect the brightness of bulb A. In practice, bulb A dims just a little when the switch closes. To see why, assume that the 1.50 V battery has an internal resistance r = 0.30 Ω and that the resistance of a glowing bulb is R...
  7. C

    Percent Change Of current when the switch is closed

    (.174A-.181A)/.181A=-3.86% but it says it wrong, and I did (.181A-.174A)/.174A =4.02% but this was wrong too. I've tried 3.87%,3.86%,-3.87%,-3.67%,4.02%, and -4.02% but all were wrong. I'm really not sure what to do here.
  8. jk22

    I Do there exist surfaces whose boundary is a closed knot?

    I ask this for the condition of application of Stoke's theorem.
  9. nomadreid

    I Quantum logic based on closed Hilbert space subspaces

    One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps...
  10. W

    B Measurement of an unknown velocity vector of a closed space

    Hi I found this paper on the measurement of unknown velocity vector of a closed space. Does it mean that it is possible to measure the unknown velocity vector of a closed space ? Can someone explain it to me
  11. Math Amateur

    I Closed Subsets in a Toplogical space ....

    I am reading Sasho Kalajdzievski's book: "An Illustrated Introduction to Topology and Homotopy" and am currently focused on Chapter 3: Topological Spaces: Definitions and Examples ... ... I need some help in order to fully understand Kalajdzievski's definition of a closed set in a...
  12. W

    Closed vs Open Electrical Circuits

    Really coming at this from a radio perspective, where texts refer to closed and open circuits as jargon; never defining. A transmitter sending waves to feed line lacking an antenna = open circuit. Check. A wire of little resistance errantly falling across circuit lines = short or closed...
  13. P

    A Parallel transport of a 1-form aound a closed loop

    Good day all. Since the gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Then If we form the Gradient vector field...
  14. V

    Water level in a submerged closed vessel after it is turned upright

    Hello all, I need Your help with one real / theoretical situation. We have submerged pipe filled with air at atmospheric pressure that is closed on both ends. The hole is formed on one pipeline end and the water starts flooding the pipe. Let's assume pipe is tilted somewhat and that no air...
  15. johnconner

    B Dilating or expanding a closed ball in Riemannian geometry

    Hello. If a closed ball is expanding in time would we say it's expanding or dilating in Riemannian geometry? better saying is I don't know which is which? and what is the function that explains the changes of coordinates of an arbitrary point on the sphere of the ball?
  16. L

    A Difference Between Subgroup & Closed Subgroup of a Group

    What is difference between subgroup and closed subgroup of the group? It is confusing to me because every group is closed. In a book Lie groups, Lie algebras and representations by Brian C. Hall is written "The condition that ##G## is closed subgroup, as opposed to merely a subgroup, should be...
  17. I

    Bernoulli equation in a closed loop system

    P1 = 5psi P2= 15psi , Z2-Z1 = 0, i assume V2 =V1 because velocity of water is the same everywhere in a pipe of constant diameter is H friction = H pump = 10psi ? Please help
  18. HRubss

    Calculating Flux over the closed surface of a cylinder

    I wanted to check my answer because I'm getting two different answers with the use of the the Divergence theorem. For the left part of the equation, I converted it so that I can evaluate the integral in polar coordinates. \oint \oint (\overrightarrow{V}\cdot\hat{n}) dS = \oint \oint...
  19. R

    B The Universe is now closed and curved?

    I read a news article about a recent study into the cmb which suggests that the universe is now been discovered to be closed and curved. Not seen anything here so am assuming it’s just another misinterpreted pop science piece. Anyone know what the article is talking about please? please see...
  20. torito_verdejo

    I Gauss' Law applicability on any closed surface

    I have read multiple threads on Physics Forums, Stackexchange and Quora, as well as the explanation of Gauss Law, but still don't understand the most fundamental aspect of it: its applicability for any kind of surface. More precisely, I don't get how this follows from the fact that...
  21. J

    I Discrete Topology and Closed Sets

    I am trying to learn some topology and was looking at a problem in the back of the book asking to show that a topological space with the property that all set are closed is a discrete space which, as understand it, means that all possible subsets are in the topology and since all subsets are...
  22. N

    MHB Infinite dimentional subspace need not be closed

    Let X=C[O,1] and Y=span($X_{0},X_{1},···$), where $X_{j}={t}^{i}$, so that Y is the set of all polynomials. Y is not closed in X.
  23. anita chandra

    A Does this integration have a closed form solution?

    I was trying to solve a differential equation that I defined to study the dynamics of a system. Meanwhile, I encounter integration. The integration is shown in the image below. I tried some solutions but I am failed to get a solution. In one solution, I took "x" common from the denominator terms...
  24. S

    I Okay, what exactly makes a timelike curve closed?

    This question has been bothering me for a long time. It is simple enough to determine whether or not a curve is timelike. You simply use this formula: gab(dxa/ds)(dxb/ds) (where x(s) is our parameterized curve). Assuming a (- + + +) signature, if the answer to the above summation is negative...
  25. MidgetDwarf

    Intro Real Analysis: Closed and Open sets Of R. Help with Problem

    For the set A: Note that if n is odd, then ## A = \{ -1 + \frac {2} {n} : \text{n is an odd integer} \} ## . If n is even, A = ## \{1 + ~ \frac {2} {n} : \text{ n is an even integer} \} ## . By a previous exercise, we know that ## \frac {1} {n} ## -> 0. Let ## A_1 ## be the sequence when n...
  26. S

    I Is this a closed spacelike curve?

    I've been refurbishing my understanding of some relativistic concepts and I've been specifically studying the concepts of spacelike, timelike and lightlike curves. According to the notes that I have been reading, curves on a Lorentzian manifold can be classified as follows: If you have a...
  27. Z

    Implications of a closed universe and recollapse

    Summary: Looking for detail about what the recollapse of a closed universe would entail As I understand it, in a universe in which the density parameter is greater than 1, a closed universe, everything would eventually recollapse upon itself. My question is: does this recollapse just refer to...
  28. F

    Is the set open, closed, neither, bounded, connected?

    Let ##z = a + bi##. Using the definition of modulus, we have ##\vert z - 3 \vert < 2## is equivalent to ##\sqrt{(a+3)^2 + b^2} < 2##. Squaring both sides we get ##(a+3)^2 + b^2 < 4##. This is the open disk center at ##3## with radius ##4## which we write as ##D[-3, 2]##. First we show...
  29. Math Amateur

    MHB Closed and Bounded Intervals are Compact .... Sohrab, Propostion 4.1.9 .... ....

    Closed and Bounded Intervals are Compact ... Sohrab, Proposition 4.1.9 ... ... I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Proposition...
  30. M

    B Measuring Fictitious Forces in a Closed Box

    I was reading the following article: https://en.wikipedia.org/wiki/Fictitious_force When I came across this passage: "This led Albert Einstein to wonder whether gravity was a fictitious force as well. He noted that a freefalling observer in a closed box would not be able to detect the force of...
  31. M

    Energy interactions of a stationary closed system

    A stationary closed system such as an air in a room or a water in tank can exchange energy with its surroundings, such as receiving heat, fan work, electrical work, shaft work. These energy interactions cause a change in the total energy E of a system. This total energy can be comprised of...
  32. Math Amateur

    MHB Open and Closed Sets in R^n .... Duistermaat and Kolk, Lemma 1.2.11 ....

    I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with the proof of Lemma 1.2.11 ... Duistermaat and Kolk"s Lemma 1.2.11 reads as follows: Can someone please demonstrate...
  33. B

    Closed form of the position of a bouncing ball

    I know that the height before the first bounce will be ##y = g * t * t + v_0 * t + y_0##. After the first bounce, I can find y by pretending the ball was thrown from the ground with velocity ##e * -v_f## with ##v_f## being the velocity of the ball when hitting the ground, but I have to reset the...
  34. Bandersnatch

    I Why is this Hubble plot linear for Omega=2 closed universe?

    On Ned Wright's pages one can find this graph: plotting some supernova data against different expansion models. The main thing here that gives me a pause is the linear relationship for the closed universe with ##\Omega##=2 (red line). There doesn't seem to be any weird scaling involved. What is...
  35. Math Amateur

    MHB Open and Closed Sets .... Conway, Example 5.3.4 (b) .... ....

    I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.3: Open and Closed Sets ... Conway's Example 5.3,4 (b) reads as follows ... ... Note that Conway defines open and closed...
  36. Math Amateur

    I Open and Closed Sets .... Conway, Example 5.3.4 (b) .... ....

    I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.3: Open and Closed Sets ... Conway's Example 5.3,4 (b) reads as follows ... ... Note that Conway defines open and closed sets...
  37. QuarkDecay

    A How Does the 3D Metric Relate to Volume in a Closed Universe?

    The final result must be V=2π2α3 Hint says we must use the dV in the spherical system (dV=r2sin2θdrdθdφ) as well as the equation of the three-dimensional metric ds2= c2dt2 - a2[ dr2/(1-kr2) +r2(dθ2 +sin2θ dφ2) ] For a closed universe we know k=+1 and with dt=0 My problem is, I don't understand...
  38. M

    A Why does the characteristic function not converge in the L1 space?

    Hi PF! When proving a closed ball in ##L_1[0,1]## is not compact, I came across a proof, which states it is enough to prove that the space is not sequentially compact. Counter example: consider the sequence of functions ##g_n:x \mapsto x^n##. The sequence is bounded as for all ##n\in \mathbb N...
  39. MedEx

    Finding the Current in a Resistor in a Closed Circuit

    1. Homework Statement The ideal battery in Figure (a) has emf = 7.7 V. Plot 1 in Figure (b) gives the electric potential difference V that can appear across resistor 1 of the circuit versus the current i in that resistor. The scale of the V axis is set by Vs = 18.9 V, and the scale of the i...
  40. Q

    A cyclist riding on a closed path....

    At the junction of paths in the park there is a closed loop in the shape of eight. Straight sections intersect at right angles, and the circular sections follow in the tangent direction. Overall the length of the loop is s=280 m and the cyclist has run through it in a uniform motion over time t=...
  41. L

    Electrical Way to generate energy in a closed system?

    I’m looking to create a self sustaining ecosystem inside of a 4x2’ glass box that could be a totally closed system (with the exception of heat transfer, I can’t really prevent that). The idea is that I would bury it for a year and then dig it up again. In order for that to work I would need a...
  42. Mr Davis 97

    I Trying to understand why a set is both open and closed

    Take the subset of ##\mathbb{R}##, ##X = [0,1]\cup [2,3]##. Under the usual metric, the set ##X## is open and closed, according to my text. How is this the case? In particular, how is ##[0,1]## open in ##X##?
  43. Mr Davis 97

    Showing that a half-open set is neither open nor closed

    Homework Statement Where ##a,b\in \mathbb{R}##, show that ##[a,b)## is not open. Homework EquationsThe Attempt at a Solution I need to show that there exists an ##x\in [a,b)## such that for all ##\epsilon > 0##, ##B_\epsilon (x) \not \subseteq [a,b)##. To this end put ##x=a##, and let...
  44. Mr Davis 97

    Infinite union of closed sets is not closed

    Homework Statement Show that it is not necessarily true that the infinite union of closed sets is closed Homework EquationsThe Attempt at a Solution From intuition, I came up with the following counter-example: ##\displaystyle \bigcup_{n=2}^{\infty} \left[ \frac{1}{n}, \frac{n}{n+1} \right] =...
  45. J

    MHB Mass of weight to keep gate closed

    Hi All, Cannot seem to figure out the question below. I`ve attached an image of the question. I basically need to find the mass of the weight W. The hinged gate will open when the water height is 12ft. The gate is a 5ft wide L shaped gate, hinged at point A. I`ve gone down the line of CW =...
  46. TheQuestionGuy14

    B Closed Timelike Curves vs Time Loops: What's the Difference?

    What's the difference between Closed Timelike Curves seen in physics and the time loops in movies where a character relives a period of time over and over, but retains memories? I'm just curious about this stuff.
  47. Mr Davis 97

    Finding a closed form expression for an infinite union

    Homework Statement Show that ##\displaystyle \bigcup_{n=2}^\infty \left[ \frac{1}{n} , \frac{n}{n+1} \right] = (0,1)##. Homework EquationsThe Attempt at a Solution I'm not sure how to show this rigorously. It is sufficient to note that ##\lim_{n\to\infty} \frac{1}{n} = 0## and that...
  48. MatthijsRog

    What constitutes a closed end in acoustic resonance in tubes

    Dear all, For my students, I'm currently trying out some experiments they can do to simulate acoustic processes. One of the topics that we will be discussing is that of standing waves. Although I have never done it before--I come from a completely different background--I want to create...
  49. MPavsic

    Induced EMF in closed loops around bar magnet

    I wonder if there is a way to calculate induced EMF in closed loops around bar magnet, which is traveling with constant velocity v to the right as depicted?
  50. Gene Naden

    I How to prove that compact regions in surfaces are closed?

    This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p. Here is my...
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