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. D

    Proving that a set, which is defined via distance to a closed set, is closed

    This was a problem on a recent graduate level introductory analysis midterm. The entire class completely bombed (class average was 18%), so we have to rewrite the exam as homework. So with those considerations, i don't want any explicit help, but feedback on the other hand would be great. I'm...
  2. A

    Is that subset of the set of continuous differential functions closed?

    Hi! I have used the physics forum a lot of times to deal with several tasks that I had and now its the time to introduce my own query! So please bear with me :-) Homework Statement Equip the set C^1_{[0,1]} with the inner product: \left\langle f,g \right\rangle= \int_{0}^{1}...
  3. A

    Reducing the entropy of a closed system

    Originally posted on sciforumsDOTcom by me (DRZion): So I came up with a scenario which is simple enough for anyone to understand. You take a fluid which is liquid at room temperature, but freezes to a become a solid denser than the liquid. This is done to any amount of liquid at the...
  4. L

    Thermodynamics closed cycle entropy cycle

    Homework Statement The figure below shows a plot of temperature T versus entropy S for the closed cycle of a particular heat engine (not necessarily an ideal gas) which consists of 3 processes and which operates between two heater reservoirs, a hot reservoir with temperature T1 and a cold...
  5. J

    Thermodynamics: closed system problem

    As shown in the figure attached, an insulated box is initially divided into halves by a frictionless, thermally conducting piston. On one side of the piston is 1.5 m^3 of air at 400 K, 4 bar. On the other side is 1.5 m^3 of air at 400 K, 2 bar. The piston is released and equilibrium is attained...
  6. S

    Uncountable family of disjoint closed sets

    Homework Statement Determine whether the following statements are true or false a) Every pairwise disjoint family of open subsets of ℝ is countable. b) Every pairwise disjoint family of closed subsets of ℝ is countable. Homework Equations part (a) is true. we can find 1-1...
  7. J

    Proof that the nullspace is closed in addition/multiplication

    Homework Statement It is number three on the following page. http://people.math.carleton.ca/~mezo/A3math1102-11.pdfHomework Equations No idea. The Attempt at a Solution I have no idea how to incorporate the kj. Best I could reason through this is supposing: b1 ∈ N(A) , c1 ∈ N(A) Ab1 +...
  8. I

    Voltage drop difference in closed and open circuits

    So the voltage drops across closed circuits I get that is P = V^2/R to get the power and then you will use P=I^2/R to get the current running through the circuit and in the case the current flowing through the closed circuit is equal in every resistor and so is the voltage drop. The part that I...
  9. F

    Non-equal Gravitational Potential and Kinetic Energy in a closed system

    While trying to get my head around Gravitational Potential Energy I devised the following simple system: Point Mass A of 1kg is 1000m away from Point Mass B of 100kg within an empty universe. The gravitational force exerted by A on B is G*10^-16; by B on A is G*10^-4. At time=0 these two...
  10. M

    Function continuous, then a subset is closed

    Homework Statement Let M, N be two metric spaces. For f: M --> N, define the function on M, graph(f) = {(x,f(x)) \inMxN: x\inM} show f continuous => graph(f) is closed in MxN Homework Equations The Attempt at a Solution I can't figure out what method to use. I have...
  11. G

    Proving Compact Sets Must Be Closed

    Homework Statement Show that every compact set must be closed. I am looking for a simple proof. This is supposed to be Intro Analysis proof. Relevant equations Any compact set must be bounded. The Attempt at a Solution Suppose A is not closed, so let a be an accumulation...
  12. rcgldr

    Show PE to KE change in closed system is independent of initial velocity

    Because of the kinetic energy and frames of reference thread: https://www.physicsforums.com/showthread.php?t=534883 I was wondering how to show that a change from potential to kinetic energy in a closed system is independent of the (inertial) frame of reference. I think the math below...
  13. M

    A set is closed iff it equals an intersection of closed sets

    Homework Statement Let M be a metric space, A a subset of M, x a point in M. Define the metric of x to A by d(x,A) = inf d(x,y), y in A For \epsilon>0, define the sets D(A,\epsilon) = {x in M : d(x,A)<\epsilon} N(A,\epsilon) = {x in M: d(x,A)\leq\epsilon} Show that A is...
  14. I

    A closed cylindrical can of fixed volume V has radius r

    (a) Find the surface area, S, as a function of r. S = 2*pi*r^2 + 2*pi*r*h I know how the 2pi(r)^2 is found, but where does the 2*pi*r*h come from? (b) What happens to the value of S as r goes to infinity? S also goes to infinity. As r increases, S increases. (c) Sketch a graph of...
  15. Q

    Help finding the midpoint of a closed interval & more

    Hey everyone, This is a review of some stuff I learned in high school, but I haven't actually done anything calculus related in about 2 years, and to be honest it looks foreign to me, if someone could help jog the old noodle it would help tremendously. The first question is as follows...
  16. A

    Prove this is a closed set/Real Analysis

    Homework Statement Prove that {(a1, a2) ∈ R2 : 0 ≤ a1 ≤ 2, 0 ≤ a2 ≤ 4} is a closed set in the Euclidean metric. Homework Equations Not sure The Attempt at a Solution How do I approach this problem? Do I prove there is a closed ball in my set? Or, do I prove there is an open ball...
  17. F

    Q is closed under division or not?

    Homework Statement Q = rational numbers My professor proved that it is closed under addition yesterday. I kinda understood a bit... How the heck does it work for when n_1 or n_2 = 0?
  18. B

    Is (2, 3) a Closed Set in the Phase Space X = [0, 1] ∪ (2, 3)?

    X = [0, 1] \bigcup (2,3) is phase space. Show that (2, 3) open and closed set of X . the question is like that but I think it is false because it is not close, right?
  19. X

    Understanding Closed and Open Sets in R^d

    This is the question: Let A be an open set and B a closed set. If B ⊂ A, prove that A \ B is open. If A ⊂ B, prove that B \ A is closed. Right before this we have a theorem stated as below: In R^d, (a) the union of an arbitrary collection of open sets is open; (b) the intersection of any...
  20. D

    CDF of a normal vector is there a closed form?

    Suppose a vector X of length n, where each component of X is normally distributed with mean 0 and variance 1, and independent of the other components. I want to know the probability that at least one of X_1>2, X_2>2.5, X_3>1.9, etc. happens (inclusive), i.e. the probability that the vector's...
  21. jaumzaum

    What Happens to the Missing Energy in a Closed Circuit with Capacitors?

    Hi everyone, I was studying eletrodynamics and I've found the the following question In ideal circuit of the figure, initially opened, the capacitor of capacitance CX is initially loaded and stores a eletric potencial energy E. The capacitor of capacitance CY = 2CX is initially...
  22. A

    Open and Closed Relations: A Topological Approach to Evaluating Limits

    "Open" and "closed" relations We know that if we have convergent sequences (xn) and (yn) in simply ordered metric space, then xn\leqyn implies that the limits x and y have x\leqy. Also, xn<yn. My instinct on noting this is to say that "<" is an "open relation" on that metric space, and that...
  23. E

    Can We Observe Our Own Galaxy's Development in its Early Stages?

    Would it be possible to observe our own Milky Way Galaxy developing in its early stages? Or at least possible for a very old galaxy to observe itself developing in it's infant stages?
  24. A

    How is the theorem of Stoke's proved for closed submanifolds without boundaries?

    Hi guys! I am reading a paper which uses closed forms \omega on a p-dimensional closed submanifold \Sigma of a larger manifold M. When we integrate \omega we get a number Q(\Sigma) =\int _{\Sigma}\omega which, in principle, depends on the choice of \Sigma but because \omega is closed...
  25. I

    Sublimation of Dry Ice in a Closed System

    Hello, Please forgive my ignorance, although bright, I was a lousy student, & never took physics in school. I find it frustrating when relatives & friends are uncertain as to how to respond to questions like those below, so your educated reply would therefore be all the more appreciated...
  26. B

    Closed sets in Cantor Space that are not Clopen

    Hi, Is there a characterization of subsets of the Cantor space C that are closed but not open? As a totally-disconnected set/space, C has a basis of clopen sets; but I'm just curious of what the closed non-open sets are.
  27. S

    (LinearAlgebra) all 2x2 invertible matrices closed under addition?

    Homework Statement Suppose V is a vector space. Is the set of all 2x2 invertible matrices closed under addition? If so, please prove it. If not, please provide a counter-example. Homework Equations The Attempt at a Solution well i know that what does it mean to be closed...
  28. A

    How Can You Derive the Formula for the Sum of the First n Integers?

    Hello, i just came accros: Sum(i) , from i=1 to i=n which apparently equals n(n+1)/2 -Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation? -Do you have any resources to offer, that...
  29. A

    Open and Closed Models in Cosmology

    Let us consider the cosmological metric: {ds}^{2}{=}{dt}^{2}{-}{[}{a}{(}{t}{)}{]}^{2}{[}\frac{{dr}^{2}}{{1}{-}{k}{r}^{2}}{+}{r}^{2}{(}{d}{\theta}^{2}{+}{sin}^{2}{(}{\theta}{)}{d}{\phi}^{2}{]} -------------- (1) For closed models k is positive We shall consider here a closed one: We write...
  30. Fredrik

    Closed Subsets and Limits of Sequences: A Topology Book Example

    Anyone have a good example of a closed subset of a topological space that isn't closed under limits of sequences?
  31. E

    Why rotational kinetic energy of a closed system is not conserved?

    A simple rotating system with no external forces acting on it carries a fixed angular momentum and an associated rotational kinetic energy. If the system changes its internal configuration, such as a spinning skater retracting or extending his/her arms, the angular momentum remains constant...
  32. D

    Open and closed sets in metric spaces

    From the definition of an open set as a set containing at least one neighborhood of each of its points, and a closed set being a set containing all its limit points, how can we show that the complement of an open set is a closed set (and vice versa)? Usually this is taken as a definition, but...
  33. L

    Is the Set of Increasing Continuous Functions on [0,1] Closed?

    Homework Statement prove that the set of continuous functions on [0,1] that are increasing is a closed set. Homework Equations The Attempt at a Solution Need to prove the complement is open. So need to prove the set of continuous functions on [0,1] that are non increasing is open...
  34. Ivan Seeking

    The 405 fwy in Los Angeles will be closed for construction work. For

    The 405 fwy in Los Angeles will be closed for construction work. For anyone who has to drive in the area, what a nightmare this will be! Last night on the Tonight Show, Jay Leno actually played a video of alternate routes...
  35. N

    Suppose i have a bucket full of water in a closed room,the water is

    suppose i have a bucket full of water in a closed room,the water is kept undisturbed for say 5 days,after those 5 days i found that the temperature of water was lower than as it was 5 days before and so it does for the room all this (for the system) isn't going in accordance with second law of...
  36. S

    How does a 'open surface' of a closed loop look like?

    im on the topic of electricity and magnetism, and came across walter lewin's lecture. i cannot visualise how the 'open surface' of this solenoid will look like is the open surface a riemann surface? or something else...
  37. W

    Examples of infinite/arbitrary unions of closed sets that remain closed.

    Hello, I am trying to think of examples of these. At the moment, I can only think of ( on R ) closed intervals being the union of single-point sets ( infinitely many, the ones inside ).. et c. I also think the cantor set is an example of this. Are there more "natural" examples? Thank you for...
  38. B

    Do Closed Timelike Curves Exist in Reality or Nature?

    do they exist in reality or in nature?
  39. alemsalem

    Is every Closed set a complete space?

    a closed set contains all its cluster (accumulation) points: points for which any open neighborhood around them no matter how small contains points from the set. a complete set contains all limit points of Cauchy sequences. which are very similar to cluster points. My question is: in a metric...
  40. L

    Proving a set in closed and nowhere dense

    Homework Statement Dn={f in C([0,1]) : there exists t in [0,1] for every h in R/{0}, abs((f(t+h)-f(t))/h) <=n} prove the Dn's are closed nowhere dense sets. A subset of some set A is closed in A if its complement is open in A. Homework Equations The Attempt at a Solution i...
  41. B

    A compact, B closed Disjoint subsets of Metric Space then d(A,B)=0

    Hi, All: Let X be a metric space and let A be a compact subset of X, B a closed subset of X. I am trying to show this implies that d(A,B)=0. Please critique my proof: First, we define d(A,B) as inf{d(a,b): a in A, b in B}. We then show that compactness of A forces the existence of a in A...
  42. O

    Prove C[0,1] is closed in B[0,1] (sup norm)

    So basically, my metric space X is the set of all bounded functions from [0,1] to the reals and the metric is defined as follows: d(f,g)=sup|f(x)-g(x)| where x belongs to [0,1]. I want to prove that the set of all discontinuous bounded functions, D[0,1] in X is open. My attempt - Start with an...
  43. S

    Boundary of closed sets (Spivak's C. on M.)

    Homework Statement I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set U=\{x\in R^n : |x|\leq 1\}. While it is evident that...
  44. M

    Proving that if X is compact, X is closed

    Here is my attempt at the proof: The statement is equivalent to the proposition that if X is not closed, X is not compact. If X is not closed, there exists a limit point p that does not belong to X. Every neighborhood of p contains infinitely many points of X, which tells us also that X is...
  45. C

    Heat Loss From a Closed Box - Experimental Help

    Hi everyone. I am currently carrying out an experiment whereby I have a closed tank of water submerged in a larger tank. The water in the smaller box is heated and I am interested in the heat loss from the tank. I have done a rough theoretical calculation based on U values, where by I obtain...
  46. M

    Calculating the Net Charge Enclosed by a Closed Surface

    Homework Statement A closed surface with dimensions a = b = 0.294 m and c = 0.3528 m is located as in the figure. The electric field throughout the region is nonuniform and given by \vec{}E = (\alpha+\beta x2)ˆı where x is in meters, \alpha = 2 N/C, and \beta = 4 N/(Cm2). See figure...
  47. M

    What is a Closed Linear Subspace?

    Hi. I'm trying to find a good definition of a closed linear subspace (as opposed to any other linear subspace), and I can't find anything concise and comprehensible. Any help will be much appreciated. P.S. I'm not great at analysis, so please try to keep it simple.
  48. N

    Sources Of Error: closed air column

    I did an experiment in class to determine the speed of sound. I used the speed of sound in air equation with the room temp at 28°C which was calculated at 348.6m/s. However when doing the experiment I got 329m/s. I just want to know some source of error that may have caused this? I have...
  49. J

    No problem, glad we could help! And welcome to the site! :)

    Let X be any infinite set. The countable closed topology is defined to be the topology having as its closed sets X and all countable subsets of X. Prove that this is indeed a topology on X. Any help would be greatly appreciated. Thanks!
  50. R

    Closed orbits in hydrogen when exposed to vertical electric field

    Hi. I am trying to find the classical turning points in semi-parabolic coordinates for the hydrogen atom when an electric field is being applied to it in the y-axis. I am reading an article for those who are interested called Classical, semiclassical, and quantum dynamics in the lithium Stark...
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