closure Definition and 162 Threads

Let $$GL(2;\mathbb{C})$$ be the complex 2x2 invertible matrices group. Let $$a$$ be an irrational number and $$G$$ be the following subgroup $$G=\Big\{ \begin{pmatrix}e^{it} & 0 \\ 0 & e^{iat} \end{pmatrix} \Big| t \in \mathbb{R} \Big\}$$ I have to show that the closure of the set $$G$$...

Let $K=\mathbb{Q}[\omega]$ where $\omega^2+\omega+1=0$ and let $R$ be the polynomial ring $K[x]$. Let $L$ be the field $K(x)[y]$ where $y$ satisfies $y^3=1+x^2$.Which is the integral closure of $R$ in $L$, why?

In Rudin we read ##diam \ \bar{S} = diam \ S##. And the ##2ε## trick is very clear. However I see how would this would work for an accumulation point of ##S## but what about an Isolated point of ##S## that is miles away from the set.

Homework Statement Prove that if S is a bounded subset of ℝ^n, then the closure of S is bounded. Homework Equations Definitions of bounded, closure, open balls, etc. The Attempt at a Solution See attached pdf.

In topology, when we say a set is closed, it means it contains all of its limit points In Algebra closure of S under * is defined as if a, b are in S then a*b is in S. Are these notations similar in any way?

I can't find the source of this statement now, but I've been trying to prove that \overline{A}\setminus\overline{B} \subseteq \overline{A\setminus B}. Now x\in\overline{A}\setminus\overline{B} means x is in every closed superset of A but there exists a closed superset of B that doesn't...

Homework Statement Define: - c0 = {(xn)n \in \ell\infty : limn → \infty xn = 0} - l0 = {(xn)n \in \ell\infty : \exists N \in the natural numbers, (xn)n = 0, n \geq N}Problem: Prove that \overline{\ell}0= c0 in \ell\infty Homework EquationsThe Attempt at a Solution I want to find the...

Hello, Let's have a group G and two subgroups A<G and B<G such that the intersection of A and B is trivial. I consider the subgroup \left\langle A^B \right\rangle which is called conjugate closure of A with respect to B, and it is the subgroup generated by the set: A^B=\{ b^{-1}ab \;|\; a\in...

This isn't really hw, just me being confused over some examples. I have 'learned' the basic definitions of neighborhood, limit point, closed, and closure but have some trouble accepting the following examples. 1. For Q in R, Q is not closed. The set of all limit points of Q is R, so its...

First a little background. I'm about to start my third year of college as a physics major. I don't really date much (only one girl briefly in high school). This is mostly because I'm pretty shy and find it hard to approach women out of the blue. I've also been told I'm decently attractive, but...

Showing that the Closure of a Connected set... Show that the Closure of a Connected set is connected. Attempt: Assume that the closure of a conncted set S is disconnected. ==> S = U \cup V is a disconnection of S. (bold for closure) ==> (S\capU) \cup (S\capV) is a disconnection of...

Homework Statement How do I show that a simple set is closed? Ex: the set of points defined by the parabola y=x^2 The Attempt at a Solution Well, a set is closed iff it contains all of its limit points. So, I want to show that this is true for the given set. I'm not exactly sure...

Homework Statement Cl(S \cup T)= Cl(S) \cup Cl(T)Homework Equations I'm using the fact that the closure of a set is equal to itself union its limit points.The Attempt at a Solution I am just having trouble with showing Cl(S \cup T) \subset Cl(S) \cup Cl(T). I can prove this one way, but I...

Hi all! I am searching for an algorithm (most likely already present in the literature) that could solve the following problem: Instance: Properties of sets of elements and relations between sets of elements Question: Find the closure of the properties and relations Possible properties...

Hello, Let me first just say, I posted this thread on mathhelpforum.com - but I read a post by Plato somewhere or another recommending here instead, since apparently the other site had some bad customer service issues... (: I want to prove that if given two functions f and g (f is assumed...

Hi, my instructor left this as an exercise, but I got confused in the second part. Could you please help me? cl(A\capB)\subseteqcl A \cap cl B But the reverse is not true. Prove this and give a counterexample on the reverse statement. My attempt: If x\in A\capB, then x\in cl(A\capB) x\in A...

For a subset which is both closed and open (clopen) does its closure equal its interior?

If the closure of a space C is connected, is C connected?

Homework Statement I want to know if the definition of σ-algebra stated below implies that every σ-algebra is closed under unions, intersections and differences (of only two members). If I assume that one of those three statements is true, I can prove the others, but I don't see how to prove...

Everything about the subring test is straightforward from the subgroup test, but the multiplicative operation of the subring, S, of ring, R, needs to be closed wrt multiplication, *. How do you prove S is closed wrt * if the only assumption about * is associativity and distributivity over...

Hi, can anyone help me ? Given Topological Spaces (metric spaces) (X, d1) and (Y,d2), show that a function f: X -> Y is continuous if and only if f(cl of A) is a subset of cl of f(A) for all A subset X1. How can i proof this ? Thank you!

Is it true? " Set of interior points of the closure of A equals the set of interior points of A. "

I'm thinking proof by contradiction but I can't seem to get anywhere.

Homework Statement Let X = R2 with the Euclidean metric and let S = {(x1, x2) : x1^2+x2^2 <1}.Prove that Closure of S ={(x1,x2):x1^2+x2^2<= 1} and that the Boundary of S= { (x1, x2) : x1^2 +x2 ^2=1 } . Homework Equations The Attempt at a Solution I was able to prove all my...

Homework Statement Let S = {(x,y): x^{2}+y^{2}<1}. Prove that \overline{S} is (that formula for the unit circle) \leq 1 and the boundary to be x^{2}+y^{2}=1. Homework Equations Boundary of S is denoted as the intersection of the closure of S and the closure of S complement. p \epsilon...

Discrete Mathematics -- Symmetric Closure Math help in Numerical Analysis, Systems of I can't seem to find the way to approach this problem. Because it has symbols I don't know how to type here, I have attached an image here instead. Please help me if you can. Any input would be greatly...

I was wondering, is S a subset of S-bar its closure? For example, if p belongs to S, does p belong to S-bar too? Does it go the other way, S-bar is a subset of S? If it is true that S is a subset of S-bar does this automatically mean that S is closed? Thanks

Homework Statement Suppose R is a relation on A and let S be the transitive closure of R. Prove that Dom(S) = Dom(R) and Ran(S) = Ran(R). *Dom() signifies the domain of the specified relation. Ran() signifies the range of the specified relation.* Homework Equations Let M = {T...

Hello, After a theorem stating that the product, sum, etc of two elements of a field extension that are algebraic over the original field are also algebraic, my course states the following result (translated into english): but later in my course it defines "the algebraic closure of F" as...

I have a question about https://www.amazon.com/dp/B002RS5IKA/?tag=pfamazon01-20 product. It is an autosampler shell vial with a polyethylene plug style needle closure. Does anyone know whether the plug on this vial will be resealable? In other words, will it be able to withstand multiple...

Homework Statement show that \rho(x,A)=\rho(x,\bar{A}), where \rho is a distance metric Homework Equations \rho(x,A)=glb\left\{\rho(x,\alpha),\alpha \in A \right\} \bar{A} is the closure of A \partial A, the boundary of an arbitrary set A is the difference between its closure and its interior...

As title: How do you relate P(F) with P(F') where F' is F's closure. Conditions: P(F)\neq 0 and F non-empty. My thought is P(F)=P(F'). Is this true?

Homework Statement Show that the outer area of S = outer area of the closure of S Homework Equations The Attempt at a Solution I don't really understand 100% the difference between the set S and the closure of S. I know the closure is S \cup \partialS (the boundary of S), but...

Homework Statement As the title states, the problem asks to prove that the closure of the set of rational numbers is equal to the set of real numbers. The problem includes the standard definition of the rationals as {p/q | q ≠ 0, p,q ∈ Z} and also states that the closure of a set X ⊂ R is...

Hi there, Could you please help me in how to prove the following : If Y is a closed linear subspace of a normed space X, then if X is complete ==> X/Y is complete. Cheers, W.

The graph of a continuous funtions (R -> R) is the subset G:={(x, f(x) | x element of R} is a subset of R^2. Prove that if f is continuous, then G is closed in R^2 (with euclidean metric). I know that continuity preserves limits, so xn -> x in X means f(xn-> y in Y. and for all A element...

A^closure = X\(X\A)^interior I am REALLY bad at proofs. I never know where to start. I only have the definitions of closure and interior. I feel like they threw us in the deep end I've written like 3pages, but mostly just pictures. interior: a is an element of A^int iff there exists r>0...

Greetings all, I'm looking at some examples in the Topology: Pure and Applied text. Looking at example 2.1 Consider A=[0,1) as a subset of R with the standard topology. Then Aint=(0,1) and Aclos=[0,1]. Can someone explain to me why the union of all open sets in A is that...

Homework Statement Prove that the closure of a proper ideal in a unital Banach algebra is a proper ideal. Homework Equations The hint is to use the result of the previous exercise: If I is an ideal in a unital normed algebra A, and I≠{0}, we have I=A \Leftrightarrow I contains 1...

Homework Statement The set S = [0,1] U {3} Homework Equations I need to say whether it is closed, open, compact, complete or connected. If it is not compact, give an example why. Same thing for completeness. If its not connected, state why not. The Attempt at a Solution I think it...

Homework Statement Find the closure, interior, boundary and limit points of the set [0,1) Homework Equations The Attempt at a Solution I think that the closure is [0,1]. I believe the interior is (0,1) and the boundary are the points 0 and 1. I think the limit point may also be...

Homework Statement the small mass m sliding without friction along the looped track is to remain on the track at all times, even at the very top of the loop of radius r. a) calculate, in terms of the given quantities, the min. release height h. if actual release height is 2h, calculate...

Hello, is anybody here, who can explain to me how to compute the integral closure of a ring (in another ring)? Example: What is the integral closure of Z in Q(sqrt(2)) ? Thank you! Bye, Brian

Homework Statement The switch in the circuit has been closed for a long time before opening at t = 0 a. Find i(t) across R3 for t>0. b. Find i(t) across R2 for t>0. I have attached the circuit. Homework Equations The Attempt at a Solution The only thing I know for this problem...

Homework Statement Let X be a topological space. Let A be a connected subset of X, show that the closure of A is connected. Note: Unlike regular method, my professor wants me to prove this using an alternative route. Homework Equations a) A discrete valued map, d: X -> D, is a map...

Given that alpha is an upper bound of a given set S of real numbers, prove that the following two conditions are equivalent: a) We have alpha=sup(S) b) We have alpha belongs to S closure I'm trying to prove this using two steps. Step one being: assume a is true, then prove b is true...

Suppose that S=[0,1)U(1,2) a) What is the set of interior points of S? I thought it was (0,2) b) Given that U is the set of interior points of S, evaluate U closure. I thought that U closure=[0,2] c) Give an example of a set S of real numbers such that if U is the set of...

I've been working on this problem and would like some help or any hints. Give an example of a nonempty set A subset of R such that A = br(A) = Lim(A) = Cl(A), where br(A) denotes the boundary points of A, Lim(A) denotes the limit points of A and Cl(A) denotes the closure of A. I've tried...

The question says: Show that if Q = [a1,b1]x...x[an,bn] is a rectangle, the Q equals the closure of Int Q. The definition of closure that I have is Cl(A) = int(A) U bd(A). So I'd like to show that Cl(int(Q)) = int(int(Q)) U bd(int(Q)). But this just seems to be obvious to me which just makes...

Prove that Cl(Q) = R in the standard topology I'm really stuck on this problem, seeing as we haven't covered limit points yet in the text and are not able to use them for this proof. Can anybody provide me with help needed for this proof? Many thanks.