# 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. ### 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 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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)...

48. ### 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...
49. ### 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...
50. ### 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.