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. PeteyCoco

    Net flux through a closed sphere

    Find the net electrical flux through a closed sphere of radius R in a uniform electric field I know that the flux is going to be 0 since there is no charge enclosed, but how would I show this mathematically? The next half of the question asks about a cylinder with sides parallel to the electric...
  2. S

    How to compute stationary distribution for martrix with more than 1 closed class

    Hi, Thank you in advance if anyone can answer this question. How any stationary distributions exists in below matrix and what are the value [ .5 0 0 .5 .25 .5 .25 0 0 0 1 0 1/6 0 0 5/6 ] Any information regarding how to compute stationary distribution in a martrix with more...
  3. L

    Basic topology proof of closed interval in R

    Let \{ [a_j, b_j]\}_{j\in J} be a set of (possibly infinitely many closed intervals in R whose intersection cannot be expressed as a disjoint union of subsets of R. Prove that \bigcup\limits_{j \in J} {\{ [{a_j},{b_j}]\} } is a closed interval in R. I don't understand how to attack this...
  4. alyafey22

    MHB Integration a long closed curve is 0

    \int_{\gamma (t) }\, f(z) dz \int_{\alpha}^{\beta} \, f(\gamma (t))\, \gamma '(t) \, dt \text{Use the substitution : } \gamma (t) = \xi \int_{\gamma (\alpha) }^{\gamma (\beta)} \, f(\xi )\, d \xi \text{If we integrate around a closed loope : }\gamma (\alpha) = \gamma(\beta)...
  5. J

    Closed form expression for sum of first m terms of a combinatorial number

    Dear Physics Forums denizens, I have a tricky problem that I hope one of you can help me with. (It's for a personal project, nothing to do with school.) I'm looking for a closed-form expression for the sum of the first through m-th terms of a combinatorial number. For those of you...
  6. R

    Continuity of a complex function defined on the union of an open and closed set

    Homework Statement (i) Let U and V be open subsets of C with a function f defined on U \cup V suppose that both restrictions, f_u \mathrm{and} f_v are continuous. Show that f is continuous. (ii) Illustrate by a specific example that this may not hold if one of the sets U, V is not open...
  7. J

    Example of a Quotient Map That Is Neither Open Nor Closed

    We are just looking for an example of a quotient map that is not open nor closed. Let π: ℝxℝ -> ℝ be a projection onto the first coordinate. Let A be the subspace of ℝxℝ consisting of all points (x,y) such that x≥0 or y=0 or both. Let q:A -> ℝ be a restriction of π. ( Note: assume that q was...
  8. R

    Expansion of the Universe - Closed Universe

    The Universe is expanding at an increasing rate, and from what I've read there is still much debate around its fate. Will it continue to expand forever, slow to a constant rate of expansion, or slow and fall back in on itself? Why can we not yet prove that the Universe will eventually fall...
  9. A

    Analysis question regarding the relationship between open and closed intervals

    Let a and b be real numbers with a<b, and let x be a real number. Suppose that for each ε>0, the number x belongs to the open interval (a-ε, b+ε). Prove that x belongs to the interval [a, b].
  10. E

    Does this differential equation have a closed form?

    I was busy doodling and basically ended up constructing this differential equation: p'(t)=c(t)p(t)-c(t-T)p(t-T) Obviously I've dealt with eq's like p'(t)=c(t)p(t) but I'm getting stuck because of the second term. Does this differential equation even have a closed form? Thanks.
  11. P

    A discrete subset of a metric space is open and closed

    Hi, If X \LARGE is a metric space and E \subset X is a discrete set then is E \LARGE open or closed or both? Here's my understanding: E \LARGE is closed relative to X \LARGE. proof: If p \subset E then by definition p \LARGE is an isolated point of E \LARGE, which implies that p...
  12. P

    Closed form of an infinitely nested radical.

    Hello. Does anybody happen to know a closed form of this infinitely nested radical? http://imageshack.us/a/img268/6544/radicals.jpg By any chance, maybe you even saw it somewhere? I haven't had too much success so far. At the moment I am so desperate that I'm even willing to try and...
  13. U

    Pressure across multiple closed feedwater heaters

    Doing a T-s and P-v diagram from a real power plant Rankine cycle. The cycle, after the condenser, goes through one pump, and then in series: gland steam condenser 4 closed feedwater heaters 1 open feedwater heater another pump 2 closed feedwater heater boiler Temperature and...
  14. U

    Multiple closed feedwater heaters in series

    Homework Statement Doing a T-s and P-v diagram from a real power plant Rankine cycle. The cycle, after the condenser, goes through one pump, and then in series: gland steam condenser 4 closed feedwater heaters 1 open feedwater heater another pump 2 closed feedwater heater boiler...
  15. G

    How to show that a trajectory is closed?

    Main question in the title. I did a group work in analytic mechanics about pendulum in rotating reference frame and stumbled upon this one: http://peer.ccsd.cnrs.fr/docs/00/50/17/84/PDF/PEER_stage2_10.1016%252Fj.ijnonlinmec.2008.03.009.pdf. Does this always hold? In my solution, there are two...
  16. DrChinese

    Fair Sampling Loophole now closed for Photons

    A team including Anton Zeilinger has performed an experiment closing the so-called "fair sampling loophole" for photons. http://arxiv.org/abs/1212.0533 Bell violation with entangled photons, free of the fair-sampling assumption Marissa Giustina, Alexandra Mech, Sven Ramelow, Bernhard...
  17. T

    Closed Dimensions: Metric Tensor, Curvature & Time

    If we assumed an empty space, but also assumed space dimensions are closed ( repeat after some distance D ), what would the metric tensor look like? Is this just equivalent to a space with a constant curvature R? If so, how does R relate to D? Would the time dimension also necessarily be...
  18. C

    Metric space proof open and closed set

    Homework Statement show the set {f: ∫f(t)dt>1(integration from 0 to 1) } is an open set in the metric space ( C[0,1],||.||∞） and if A is the subset of C[0,1] defined by A={f:0<=f<=1} is closed in the norm ||.||∞ norm. Homework Equations C[0,1] is f is continuous from 0 to 1.and ||.||∞...
  19. I

    General simple closed curve equation

    I have a list of points on a 2D simple closed curve and I'd like to approximate that curve using a polynomial such that the approximation will be given by: Ʃai,jxiyj = 0 However, I still need to limit the ai,j to make sure the approximation is also a simple closed curve, while still keeping...
  20. M

    What is the Effect of Heat on Alpha Particles in a Closed Box?

    Ok, so I am currently building a cloud chamber for a major presentation in the area of Thermodynamics (or Energy and Temperature) and I've been thinking quite a bit about this theory...and i could be wrong...so don't tell me off about it... So the general physics of a cloud chamber, Isopropyl...
  21. S

    Dropping a closed loop into a magbetic field

    Homework Statement The closed loop is vertical and the magnetic field is a north and south pole such that the looped end faces the magnetic poles. A multiple choice question answer stated that the closed loop will have a smaller acceleration than 10mls ^2 as it would experience an...
  22. C

    [Differential Geometry of Curves] Regular Closed Curve function

    Let m : [0,L] → ℝ2 be a positively oriented C1 regular Jordan curve parametrized with arc length. Consider the function F : [a,b] x [a,b] → ℝ defined by F(u,v) = (1/2) ||m(u) - m(v)||2 Define a local diameter of m as the line segment between two points p = m(u) and q = m(v) such that: The...
  23. C

    [Differential Geometry of Curves] Regular Closed Curve function

    Homework Statement Let m : [0,L] --> ℝ2 be a C2 regular closed curve parametrized with arc length, and define, for an integer n > 0 and scalar ε > 2 μ(u) = m(u) + εsin(2nπu/L)Nm(u) where Nm is the unit normal to m (1) Determine a maximum ε0 such that μ is a closed regular curve for...
  24. T

    Closed and bounded in relation to compact

    So this is more so a general question and not a specific problem. What exactly is the diefference between closed and boundedness? So the definition of closed is a set that contains its interior and boundary points, and the definition of bounded is if all the numbers say in a sequence are...
  25. J

    Fintie point sets in a Hausdorff space are closed.

    This may seem like a silly question, but I'll ask it anyways. :) In the Munkres text, he proves this by showing that one-point sets are closed, which I completely understand why it follows that finite point sets are closed. He does so by showing that the arbitrary one-point set {x0} equals...
  26. C

    Oscillation of a closed subinterval

    Given \epsilon > 0 , suppose \omega_f(x) < \epsilon for each x \in [a,b] . Then show there is \delta > 0 such that for every closed interval I \in [a,b] with l(I)< \delta we have \omega_f(I) < \epsilon . My first approach to this was trying to think of it as an anaglous to the definition...
  27. B

    A limit of an indexed integral at closed interval

    Let ρ(x) be a continuous function on ℝ, which evaluates ρ(x)=0 when |x|≥1 and that meets following. ∫[-1,1]ρ(x)dx=1 And let ψ(x) be a continuous function on interval [-1,1], prove lim[n→∞] n∫[-1,1]ρ(nx)ψ(x)dx = ψ(0). is denoted. This is NOT a homework but a past exam problem of...
  28. srfriggen

    Does entropy in a closed system always increase OR remain constant (

    Does entropy in a closed system always increase OR remain constant ( in equilibrium ). I have a friend arguing it is ALWAYS increasing. His latest argument was, "if no energy enters or leaves an isolated system, the availability of the remaining energy decreases."
  29. M

    One kg of air is heated in a closed rigid vessel

    Homework Statement One kg of air is heated in a closed rigid vessel such its temperature changes from 27 C to 427 C . Find the heat transfered and change in internal energy . Assume : - R= 0.287 KJ/kg k Cv = 0786 KJ/Kg k Homework Equations Q = mXCX(T2 - T1 ) The Attempt at a...
  30. D

    Transistor acting as a switch in closed circuit (with a thermistor)

    Homework Statement Ok so I have been given a diagram which if it worked should be attached. I am having a fair bit of trouble understanding it. Now I am aware that there is both electrical and conventional currents but I am confused as to whether this is electrical or conventional. I have been...
  31. G

    Is {(x,y)∈R^2 | 2x+y≤2, x-y>4} an Open Subset of R^2?

    1. {(x,y)\in R^2 such that 2x+y<=2, x-y>4} Determine whether this subset of R^2 is open, closed or neither open nor closed. 2. I think this is an open subset but not sure how to prove it. I have rearranged the equations to give x>2, y<=-2x+1, y< x-1. I think it is open because x can get...
  32. D

    The circle as a set closed and bounded

    Hi guys, I would like to understand why a circle (and in general a n-sphere) as a subset of R^2 (in general R^(n+1)) with the standard topolgy is considered a closed and a bounded set. I think that this can be a closed set because its complement (the interior of the circle and the rest of...
  33. L

    Developing Recurrence Formula (Closed)

    Homework Statement Given binary string of length n. substrings of 1's should be even. (given 0 is a delimeter) eg) 10111 is broken down into 1 and 111 (all odd) so, for example string with H(n = 4) 0000 1100 0110 0011 1111 there are 5 of them. H(n = 3) 000 110 011 there are 3 of...
  34. D

    Closed form of harmonic series

    HI! I was wandering if there is a proof that the harmonic sum \sum\frac{1}{k} has no closed form. Something like the proof that an equation with degree more than 4 has no solution in terms of radicals.
  35. O

    MHB Are all countable sets closed?

    Hello everyone! I want to show that all countable sets are closed. I can show that finite sets are closed, and the set of all natural numbers is closed by showing its complement to be a union of open sets. Now, can I start like this: A is a countable set. Every element in A can be "mapped" to...
  36. B

    Continuity and differentiability over a closed interval

    Homework Statement http://i.imgur.com/69BmR.jpg Homework Equations The Attempt at a Solution a, c are right because f(c) is continuous. b, d are right because f'(c) is differentiable over the interval I am not sure about e. Can anyone explain to me?
  37. Y

    Showing A Compact Interval Is Closed

    Homework Statement Basically, prove the Extreme Value Theorem. "If f is a continuous function over the interval [a,b] then f reaches a max and a min on that interval." Homework Equations In this case they're more like definitions and things I have proved so far. Intervals are...
  38. G

    Show subspace of normed vector is closed under sup norm.

    http://imageshack.us/a/img141/4963/92113198.jpg hey, I'm having some trouble with this question, For part a) I know that in order for c_0 to be closed every sequence in c_0 must converge to a limit in c_0 but I am having trouble actually showing that formally with the use of the norm...
  39. F

    Closed or open question about the Extended Real line

    This is just a quick question about sets that include plus or minus infinity on the extended real line. I am wondering about this in regards to measure in analysis so specifically, is [-∞,a) open or closed? I hadn't seen the extended reals before this class and we really didn't spend anytime...
  40. G

    Understanding Pressure Distribution in a Rotating Vessel of Water

    A closed vessel full of water is rotating with constant angular velocity \Omega about a horizontal axis. Show that the surfaces of equal pressure are circular cylinders whose common axis is at a height g/{\Omega}^2 above the axis of rotation. Any ideas? I do not know how to start.
  41. W

    Is Sample Space an Open or Closed Set?

    Hi all, is sample space an open set or a closed set? Thanks in advance
  42. K

    What is the Definition of Closed Sets in Topology?

    Good day! Im currently reading the book of Steven R. Lay's "Analysis with an Introduction to Proof, 3rd ed.". According to his book, if a subset S of ℝ contains all of its boundary then it is closed. But i find this wrong since if we consider S={xεQ;0≤x≤2}, then it can be shown that S...
  43. A

    Instruments and Imaging: How to prevent condensation inside a closed system?

    Dewing of a mirror or object glass can be countered with various devices and methods, but its occurrence inside a closed system is particularly problematic. I have received the following enquiry (from an experienced observer who has used several instruments over many years) concerning...
  44. D

    MHB Closed form solution for large eigenvalues

    The eigenvalues are found by $$ \tan\lambda_n = \frac{1}{\lambda_n} $$ For large eigenvalues, the intersection get closer and closer to $\lambda_n = \pi k$ where $k\in\mathbb{Z}^+$ and $k > 15$. Is this correct? Without arbitrary picking a $k$, is there a better way to determine a $k$ for when...
  45. Ranger Mike

    How Can Closed Loop Drive Systems Achieve Precise 1000 mm Movements?

    Hello, I have a few questions regarding machine tool drive and feed back system. Say I have a 1 meter long guide way and a carriage that is driven by DC motor. This carriage glides on air bearings. The drive mechanism has pinch rollers that pinch a drive rod and these rollers are driven by...
  46. M

    How to prove something is closed and bounded, ie compact

    Homework Statement I need to prove that a closed ball(radius r about x0) is closed and bounded. The same goes for a sphere(radius r about x0). Homework Equations The Attempt at a Solution How does one go about proving something is closed and bounded? My book is not very helpful...
  47. H

    Determine if a Set is Open or Closed

    Homework Statement Determine each of the following sets as open, closed, neither or both. a) {1/n : n \in N } b) N c) Q d) \bigcap^{∞}_{n=1}(0,1/n) e) {x: |x-5|\leq 1/2 f) {x: x^2>0} Homework Equations Open sets are sets that do not contain their boundary points. Closed sets contain...
  48. M

    Spivak's proof of A closed bounded subset of R^n is compact

    Spivak's proof of "A closed bounded subset of R^n is compact" Hi guys, I'm currently taking a differential geometry course and decided I would read Spivak's Calculus on Manifolds, and then move on to his Differential Geometry series. There's a proof in here that feels unjustified to me, so...
  49. D

    Closed and open at the same time?

    Closed and open at the same time??! I am doing a reading course with a professor using Rudin's Real and Complex Analysis. I had never seen any topology before so the professor told me to work through the first two chapters of Hocking's Topology. I am taking my sweet time with it because I...
  50. D

    MHB Closed sets intersection of countable open sets

    Prove that every closed set in $\mathbb{R}$ is the intersection of a countable collection of open sets. Let $G_n$ be a countable collection of open sets. Then we would have 2 cases either $x\in\bigcap G_n$ which is a point which is closed. Or we could $(a,b)$ in all $G_n$ but how to show that...
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