What is closure: Definition and 169 Discussions

Google Closure Tools is a set of tools to help developers build rich web applications with JavaScript. It was developed by Google for use in their web applications such as Gmail, Google Docs and Google Maps.

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  1. P

    I Proof for subgroup -- How prove it is a subgroup of Z^m?

    Hi together! Say we have ## \Lambda_q{(A)} = \{\mathbf{x} \in \mathbb{Z}^m: \mathbf{x} = A^T\mathbf{s} \text{ mod }q \text{ for some } \mathbf{s} \in \mathbb{Z}^n_q\} ##. How can we proof that this is a subgroup of ##\mathbb{Z}^m## ? For a sufficient proof we need to check, closure...
  2. B

    B Are there two kinds of inverse with respect to closure?

    For every instance of addition or multiplication there is an inverse, closed on the naturals. Not every instance of subtraction and division is defined, so not closed on the naturals. This looks like two kinds of inverse. Instance inverse - the inverse of instances of addition and...
  3. C

    I In Euclidian space, closed ball is equal to closure of open ball

    Problem: Let ## (X,d) ## be a metric space, denote as ## B(c,r) = \{ x \in X : d(c,x) < r \} ## the open ball at radius ## r>0 ## around ## c \in X ##, denote as ## \bar{B}(c, r) = \{ x \in X : d(c,x) \leq r \} ## the closed ball and for all ## A \subset X ## we'll denote as ## cl(A) ## the...
  4. T

    Dynamics of Four Bar Link using Hamilton's Principle with Loop Closure

    Summary:: Can someone point me to an example solution? Hello The attached figure is a four bar link. Each of the four bars has geometry, mass, moment of inertia, etc. A torque motor drives the first link. I am looking for an example (a simple solution so I can ground my self before...
  5. Euge

    MHB Projection Map $X \times Y$: Closure Property

    Here is this week's POTW: ----- Let $X$ and $Y$ be topological spaces. If $Y$ is compact, show that the projection map $p_X : X \times Y \to X$ is closed. -----
  6. P

    Pressure change in pipe due to sudden closure of valve

    Water is flowing in the pipe with velocity v0. Upon sudden closure of the valve at T, a pressure wave travels in the -ve x direction with speed c. The task is to find ##\alpha##, where ##\Delta P = \rho_0 c (\Delta v) \alpha##. The 1st step is to set up an equation using conservation of mass...
  7. P

    I Closure in the subspace of linear combinations of vectors

    This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the...
  8. Math Amateur

    I Closure & Interior as Dual Notions .... Proving Willard Theorem 3.11 ...

    I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to prove Theorem 3.11 Part 1-a using the duality relations between closure and interior ... ..The...
  9. Math Amateur

    I Interior and Closure in a Topological Space .... .... remark by Willard

    I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to fully understand a result or formula given by Willard concerning a link between...
  10. Math Amateur

    I Closure in a Topological Space .... Willard, Theorem 3.7 .... ....

    I am reading Stephen Willard: General Topology ... ... and am currently reading Chapter 2: Topological Spaces and am currently focused on Section 1: Fundamental Concepts ... ... I need help in order to fully understand an aspect of the proof of Theorem 3.7 ... ..Theorem 3.7 and its proof...
  11. Math Amateur

    I Limit Points & Closure in a Topological Space .... Singh, Theorem 1.3.7

    I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.2: Topological Spaces ... I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to...
  12. V

    A Closure of constant function 1 on the complex set

    I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...
  13. Math Amateur

    MHB Understanding Topology: Closure, Boundary & Open/Closed Sets

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ... I need some help in order to fully understand a statement by Browder in Section 6.1 ... ... The...
  14. Mr Davis 97

    Showing that the the closure of a closure is just closure

    Homework Statement Let ##M## be a metric space. Prove that ##\overline{\overline{S}} = \overline{S}## for ##S\subseteq M##. Homework EquationsThe Attempt at a Solution First we know that ##\overline{S} \subseteq \overline{\overline{S}}## is true (just take this for granted, since I know how to...
  15. E

    Proving closure of square integrable functions.

    I'm trying to prove that the set of all square integrable functions f(x) for which ∫ab |f(x)|^2 dx is finite is a vector space. Everything but the proof of closure is trivial. To prove closure, obviously we should expand out |f(x)+g(x)|^2, which turns our integral into one of |f(x)|^2 (finite)...
  16. Mr Davis 97

    Finding the closure of some metric spaces

    Homework Statement Identify ##\bar{c}##, ##\bar{c_0}## and ##\bar{c_{00}}## in the metric spaces ##(\ell^\infty,d_\infty)##. Homework Equations The ##\ell^\infty## sequence space is $$ \ell^\infty:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\...
  17. R

    Are my definitions of interior and closure correct?

    Homework Statement Define the interior A◦ and the closure A¯ of a subset of X. Show that x ∈ A◦ if and only if there exists ε > 0 such that B(x,ε) ⊂ A.The Attempt at a Solution [/B]
  18. Math Amateur

    MHB The Closure of a Set is Closed .... Lemma 1.2.10

    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 an aspect of the proof of Lemma 1.2.10 ... Duistermaat and Kolk"s proof of Lemma 1.2.10 (including D&K's definition of a...
  19. M

    I If A is an algebra, then its uniform closure is an algebra.

    Let me give some context. Let X be a compact metric space and ##C(X)## be the set of all continuous functions ##X \to \mathbb{R}##, equipped with the uniform norm, i.e. the norm defined by ##\Vert f \Vert = \sup_{x \in X} |f(x)|## Note that this is well defined by compactness. Then, for a...
  20. B

    Closure of Connected Space is Connected

    Homework Statement If ##C## is a connected space in some topological space ##X##, then the closure ##\overline{C}## is connected. Homework EquationsThe Attempt at a Solution Suppose that ##\overline{C} = A \cup B## is separation; hence, ##A## and ##B## are disjoint and do not share limit...
  21. B

    Path-Connected Sets and Their Closures: A Counterexample?

    Homework Statement I am trying to determine whether the closure of a path-connected set is path-connected. Homework EquationsThe Attempt at a Solution Let ##S = \{(x, \sin(1/x) ~|~ x \in (0,1] \}##. Then the the closure of ##S## is the Topologist's Sine Curve, which is known not to be...
  22. T

    I Interior and closure in non-Euclidean topology

    Hello everyone, I was wondering if someone could assist me with the following problem: Let T be the topology on R generated by the topological basis B: B = {{0}, (a,b], [c,d)} a < b </ 0 0 </ c < d Compute the interior and closure of the set A: A = (−3, −2] ∪ (−1, 0) ∪ (0, 1) ∪ (2, 3) I...
  23. M

    MHB Show That $C$ Is Algebraic Closure for $F$

    Hey! :o Let $E/F$ be an algebraic extension and $C$ the algebraic closure of $E$. I want to show that the field $C$ is the algebraic closuree also for $F$. We have that $C=\{c\in E\mid c \text{ algebraic over } E\}$, i.e., every polynomial $f(x)\in E[x]$ splits completely in $C$. Since...
  24. caffeinemachine

    MHB Algebraic Closure of Fp [Lang, Algebra, Chapter 6, Problem 22]

    Problem. Let $K$ be the field obtained from $\mathbf F_p$ by adjoining all primitive $\ell$-th roots of unity for primes $\ell\neq p$. Then $K$ is algebraically closed. It suffices to show that the polynomial $x^{p^n}-x$ splits in $K$ for all $n$. In order to show this, it in turn suffices to...
  25. C

    Show that E and the closure of E have the same Jordan outer measure

    Homework Statement Let E be a bounded set in R^n Show that E and the closure of E have the same Jordan outer measure Homework Equations Jordan outer measure is defined as m^* J(E)=inf(m(b)) where B \supset E B is elementary. 3. The Attempt at a Solution [/B] If E and the closure...
  26. caffeinemachine

    MHB Finite Extension Simple iff Purely Inseparable Closure Simple

    Question. Is it true that a finite extension $K:F$ is simple iff the purely inseprable closure is simple over $F$? I think have an argument to support the above. First we show the following: Lemma. Let $K:F$ be a finite extension and $S$ and $I$ be the separable and purely inseparable...
  27. A

    MHB Sigma Algebra: Seeking Help on Closure & Countable Unions

    Hi everyone, didn't know where to post question on sigma algebra so here it is:- What I've tried till now: Let C\in G 1) For C=X, f^{-1}(B)=X which will be true for B=Y (by definition) 2) For closure under complementation, to show C^{c}\in G. So, C^{c}=X\setminus C=X\setminus...
  28. R

    A Valve closure boundary condition

    does anyone know what the boundary condition is for a closing valve using the wave equation pde?
  29. L

    I Closure Phase - Interferometry - Recurrence Relation

    Hello, I'm trying to calculate a recurrence relation of the phases of 3 telescopes in a closure phase. Usually in a stellar interferometer we have 3 telescopes, located in a triangle, measuring intensity of light in 3 points on a far field plane. I found an article, describing how the phase is...
  30. I

    Limit points, closure of set (Is my proof correct?)

    Homework Statement Let ##E'## be the set of all limit points of a set ##E##. Prove that ##E'## is closed. Prove that ##E## and ##\bar E = E \cup E'## have the same limit points. Do ##E## and ##E'## always have the same limit points? Homework Equations Theorem: (i) ##\bar E## is closed (ii)...
  31. Z

    MHB Closure under Scalar multiplication

    Hey guys, I really need help in revising my Axiom 6 for my Linear Algebra course. My professor said, "You need to refine your statement. You want to show rx1 and rx2 are real numbers. You should not state they are real numbers." Here is my work: Proof of Axiom 6: rX is in R2 for X in R2...
  32. I

    Closure of Open Ball in Normed Vector Spaces

    Homework Statement Show that ##\overline{B(a,r)} = \{ x \in \mathbf R^n ; |a-x| \le r \}## in ##\mathbf R^n## for all points ##a \in \mathbf R^n## and ##r>0##. Is it possible to generalize the statement to any normed vector space? Give a example of a metric space where the statement is not...
  33. amjad-sh

    Closure relation in infinite dimensions

    The closure relation in infinite dimension is : ∫|x><x|dx =I (identity operator),but if we apply the limit definition of the integral the result is not logic or intuitive. The limit definition of the integral is a∫b f(x)dx=lim(n-->∞) [i=1]∑[i=∞]f(ci)Δxi, where Δxi=(b-a)/n (n--.>∞) and...
  34. M

    Building a Faraday Cage: Is 2-Layer Aluminum Screen Enough?

    I am building a walk in size faraday cage. In order to make it re-positionable, I am creating 6 wooden frames (top, bottom and sides) covered in 2 or 3 layers of aluminum screen from a hardware store. I am using at least 2 layers of screen, because that is how many layers it takes to cause my...
  35. PsychonautQQ

    Algebraic Closure: Finite Fields & Equivalent Statements

    I'm confused on why exactly the following two statements are equivalent for a finite field K: -If K has no proper finite extensions, then K is algebraically closed. -If every irreducible polynomial p with coefficients in K is linear then K is closed. Can somebody help shed some light on this?
  36. C

    When is integral closure generated by one element

    Hello, This is not a homework problem, nor a textbook question. Please do not remove. Is there a concrete example of the following setup : R is an integrally closed domain, a is an integral element over R, S is the integral closure of R[a] in its fraction field, S is not of the form R{[}b{]} for...
  37. C

    Quotient field of the integral closure of a ring

    This is probably a stupid question. Let R be a domain, K its field of fractions, L a finite (say) extension of K, and S the integral closure of R in L. Is the quotient field of S equal to L ? I believe that not, but I have no counter-example.
  38. M

    Topology on a set ##X## (find interior, closure and boundary of sets)

    Homework Statement . Let ##X## be a nonempty set and let ##x_0 \in X##. (a) ##\{U \in \mathcal P(X) : x_0 \in U\} \cup \{\emptyset\}## is a topology on ##X##. (b) ##\{U \in \mathcal P(X) : x_0 \not \in U\} \cup \{X\}## is a topology on ##X##. Describe the interior, the closure and the...
  39. C

    Proof of equality of diameter of a set and its closure

    In showing diam(cl(A)) ≤ diam(A), (cl(A)=closure of A) one method of proof* involves letting x,y be points in cl(A) and saying that for any radius r>0, balls B(x,r) and B(y,r) exist such that the balls intersect with A. But if x,y is in cl(A), isn't there the possibility that x,y are...
  40. A

    Determine the interior, the boundary and the closure of the set

    Homework Statement Determine the interior, the boundary and the closure of the set {z ε: Re(z2>1} Is the interior of the set path-connected? Homework Equations Re(z)=(z+z*)/2 The Attempt at a Solution Alright so z2=(x+iy)(x+iy)=x2+2ixy-y2 so Re(x2+2ixy-y2)= x2-y2 >1 So would...
  41. A

    Describe the closure of the set with formulas

    Homework Statement -∏<arg(z)<∏ (z≠0) Homework Equations arg(z) is the angle from y=0The Attempt at a Solution Arg(z) spans the entire graph since -pi to pi is the full 360 degrees so I put: -∏<arg(z)<∏ --> 0<arg(z)<2∏+k∏, (k ε Z) --> arg(z) \subset R --> arg(z) = R: all real numbers but I...
  42. D

    Proof for Closure of Vector Addition - Can You Help?

    If the tangent space at p is a true vector space, then it must be that the sum of two vectors is itself a directional derivative operator along some path passing through p. I've been trying to prove that this is true without any luck. My textbook "proves" that vector addition is closed by...
  43. M

    Finding 8 Relations on a Set of 3 Elements with the Same Symmetric Closure

    Homework Statement Show that if a set has 3 elements, then we can find 8 relations on A that all have the same symmetric closure. Homework Equations Symmetric closure ##R^* = R \cup R^{-1} ## The Attempt at a Solution If the symmetric closures of n relations are the same then...
  44. B

    Is the closure of a set the same as its smallest closed set containing it?

    My first analysis/topology text defined the boundary of a set S as the set of all points whose neighborhoods had some point in the set S and some point outside the set S. It also defined the closure of a set S the union of S and its boundary. Using this, we can prove that the closure of S is...
  45. stripes

    Searching for Closure: Mult. Inverses & Addition

    Homework Statement Homework Equations The Attempt at a Solution Well thankfully I just have to present closure under mult. inverses and closure under addition. But I seem to be going in circles...if a is in G, then we need to show that a-1 is also in G. So a*a-1 = 1F, but is...
  46. Mandelbroth

    Proof Check: Closure of Union Contains Union of Closures

    I intend to show, for a set ##X## containing ##A_i## for all ##i##, $$\overline{\bigcup A_i}\supseteq \bigcup \overline{A_i}.$$ //Proof: We proceed to prove that ##\forall x\in X,~x\in\bigcup\overline{A_i}\implies x\in\overline{\bigcup A_i}##. Equivalently, ##\forall x\in...
  47. B

    Closure in Groups: Definition & Examples

    Let G be a group and my book defines closure as: For all a,bε G the element a*b is a well defined element of G. Then G is called a group. When they say well defined element does that mean I have to show a*b is well defined and it is a element of the group? Or do I just show a*b is closed under...
  48. mnb96

    Question on conjugation closure of subgroups

    Hello, I consider the groups of rotations R=SO(2) and the group T of translations on the 2D Cartesian plane. Let's define Ω as the group Ω=RT. Thus Ω is essentially SE(2), the special Euclidean group. It is known that R and T are respectively 1-dimensional and 2-dimensional Lie groups...
  49. A

    The closure of an open set A, strictly bigger than A itself?

    Hi there! Is the following true? Suppose A is an open set and not closed. Cl(A) is closed and contains A, hence it contains at least one point not in A. If A is both open and closed it obviously does not hold.