closure Definition and 162 Threads

Homework Statement See attachment Homework Equations The Attempt at a Solution I am not sure how I should approach this first off. I have tried this 3 ways but I always decide they don't work. Click on the other attachment to see my work, It's only the first part of the first...

Homework Statement Prove or disprove the following statement: The closure of a set S is closed. Homework Equations Definition of closure: set T is the closure of set S means that T is the union of S and the set of limit points of S. Definition of a closed set: set S is...

Homework Statement Let A = {ln(1 + q^2) : q is rational}. One needs to find Cl(A) in R with its euclidean topology. The Attempt at a Solution So, the set A is a countable subset of [0, +∞>. The closure is, by definition, the intersection of all closed sets containing A. So, Cl(A) would be...

Homework Statement Let X be a topological space. If A is a subset of X, the the boundary of A is closure(A) intersect closure(X-A). a. prove that interior(A) and boundary(A) are disjoint and that closure(A)=interior(A) union boundary(A) b. prove that U is open iff Boundary(U)=closure(U)-U...

I'm not sure about my answers, any help is highly appreciated. Let (N, U) be a topological space, where N is the set of natural numbers (without 0), and U = {0} U {Oi, i is from N}, where Oi = {i, i+1, i+2, ...} and {0} is the empty set. One has to find the interior (Int) and closure (Cl) of...

Why is it that \mathbb{C}(x) (\mathbb{C} adjoined with x) is not algebraically closed? Here x is an indeterminate. My first question is what does the field extension \mathbb{C}(x) even mean? If E is a field extension of F, and a is an transcendental element of E over F, then the notation...

I'm reading about a theorem that has as an assumption that the closure of some one-parameter subgroup is a torus. Could someone provide an example of a case where the closure of a one-parameter subgroup is of dimension greater than 1? Thanks.

Over the last few weeks, I have been participating in a few discussions with tea party members. Although we never came to any agreements (Apparently, I'm a liberal communist Nazi freedom-hating socialist), there was an interesting trend to our discussions. In any discussion of sufficient...

Homework Statement What is the closure of F = { (x,sin(1/x) : 0<x<=1 }? Homework Equations None The Attempt at a Solution F is a squiggly line in R2. For every point in F (every point on the squiggly line) an open ball about that point will contain point both in F and in the...

Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F. Claim 1: F is contained in the clousre of F...

Homework Statement Let (X,d) be a metric space and E is a subset of X. Prove that (c means complement, E bar means the closure of E) Homework Equations N/A The Attempt at a Solution Let (X,d) be a metric space and B(r,x) is the open ball of radius r about x. Definition: Let F be...

Homework Statement Let W\subset S \subset \mathbb{R}^n. Show that the following are equivalent: (i) W is relatively closed in S, (ii) W = \bar{W}\cap S and (iii) (\partial W)\cap S \subset W. Homework Equations The only thing we have to work with is the definitions of open and closed sets...

Ok, this is a really easy question, so I apologise in advance. Let A be an abelian subgroup of a topological group. I want to show that cl(A) is also. Now I've shown that cl(A) is a subgroup, that is fairly easy. So I just need to show it is abelian. For a metric space, it is easy...

Homework Statement Let S = {1, 2, 3, 4}. For each of the following relationson S give its transitive closure. (a) {(1, 1), (3, 4)} (b) {(1, 2), (4, 4), (2, 1), (4, 3), (2, 3)} (c) {(1, 1), (2, 2), (3, 3), (4, 4), (4, 1)} (d) {(1, 3), (3, 2), (2, 4), (4, 1)} Homework Equations N/A...

Dear all, How can I show that: The boundary of a set S is equal to the intersection of the closure and the closure of the complement of S ? Thanks a lot in advance

Homework Statement determine if the space is a subspace testing both closure axioms. in R^2 the set of vectors (a,b) where ab=0 Homework Equations The Attempt at a Solution i just used the sum and product which are the closure axioms. But at the end how do you tell if the...

Given that f is a function from R(=real Nos) to R continuous on R AND ,A any subset of R,IS THE closure of f(A) ,a closed set??

Hello Everyone, first of all my apologies, may be my question is too stupid for a forum on Topology and Geometry, but it's something I was thinking about for a while without getting an answer : What's the actual difference between a Close set and a Complete set? I mean : from an algebraic...

Homework Statement Let R be a relation and define the following sequence R^0 = R R^{i+1} = R^i \cup \{(s,u) \vert \exists t, (s,t) \in R^i, (t,u) \in R^i \} And R^{+} = \bigcup_{i} R_i Prove that R^{+} is the transitive closure. Homework Equations The Attempt at a...

Homework Statement 1) Let S be a set and p: SxS->S be a binary operation. If T is a subset of S, then T is closed under p if p: TxT->T. As an example let S = integers and T be even Integers, and p be ordinary addition. Under which operations +,-,*,/ is the set Q closed? Under which...

I recently came across the following remark in a book: "Notice that a constructible set contains a dense open subset of its closure." Now this doesn't seem at all obvious to me. Let us recall the definitions first. A locally closed set is the intersection of a closed and an open set. A...

Homework Statement 6) Prove or give a counter-example of the following statements (i) (interiorA)(closure) intersect interior(A(closure)): (ii) interior(A(closure)) intersect (interiorA(closure)): (iii) interior(A union B) = interiorA union interiorB: (iv) interior(A intersect B) =...

Prove or disprove Closure of the Interior of a closed set X is equal to X so clos(intX)=X I think it is true, but i don't know how to prove it I thought that clos(int(X))=int(X)+bdy(int(X))=X thanks, julia

Homework Statement Let E be an extension field of a field F. Given \alpha\in E, show that, if \alpha\notin F_E, then \alpha is transcendental over F_E. Homework Equations F_E denotes the algebraic closure of F in its extension field E. The Attempt at a Solution First, I assumed \alpha...

Homework Statement If A is a discrete subset of the reals, prove that A'=cl_x A \backslash A is a closed set. Homework Equations A' = the derived set of A x is a derived pt of A if U \cap (A \backslash \{x\}) \neq \emptyset for every open U such that x is in U. Thrm1. A...

Homework Statement I am looking for examples of sets that have derived pts that are different from closure pts because I am trying to understand them better. Also, if you can , please try to bring the word "base" into this. I do not understand quite fully a base. I know the definition and...

hey Im having problem about closure rule can anyone explain the closure rule? why does it gives one mads

Homework Statement Let S be a set in R^n, is it true that every interior point of 'the closure of S' is in Int S? Justify. 2. Relevant theorem S^int = {x belongs to S: B(r,x) belongs to S for some r>0} The closure of S is the union of S and all its bdary points. The Attempt at...

Let S be a set in Rn, is it true that every interior point in the closure of S is in the interior of S? Justify. ie. int(closure(S)) a subset of int(S) It seems to me that it would be true...if you could say that the interior of the closure of S is the interior of S unioned with the...

can anyone give me a precise statement of the "closure of modules" theorem in several complex variables? it says something like: a criterion for the germ of a function to belong to the stalk of an ideal at a point, is that the function can be uniformly approximated on neighborhoods of that point...

Homework Statement Sketch the closure of the set:Re(1/z)=< 1/2[ b]2. Homework Equations [/b] The Attempt at a Solution Re(1/z)=Re(1/(x+iy) Re(1/(x+iy))=< 1/2. Not really sure how to sketch 1/2 on a complex plane. Maybe 1/2 can be written in a complex form: 1/2= (1/2)+(0)*i=1/2 and...

Let G be a finite nonempty set with an operation * such that: 1. G is closed under *. 2. * is associative. 3. Given with a*b=a*c, then b=c 4. Given with b*a=c*a, then b=c Give an example to show that under the conditions above, G is no longer a group if G is an infinite set?

Homework Statement Find the Galois closure of the field \mathbb{Q}(\alpha) over \mathbb{Q}, where \alpha = \sqrt{1 + \sqrt{2}}. Homework Equations Um...the Galois closure of E over F, where E is a finite separable extension is a Galois extension of F containing E which is minimal...

Homework Statement Does anyone know a generic way of showing that a field is closed under multiplication and addition? Please, thanks Homework Equations The Attempt at a Solution Just need to prove that a+b and ab are in the field that each element a and b are from. Any ideas??

1. Show that if a, b \in \textbf{Q}, then ab and a + b are elements of \textbf{Q} as well. 2. Show that if a \in \textbf{Q} and t \in \textbf{Q}, then a + t \in \textbf{I} and at \in \textbg{I} as long as a \neq 0. I'm just a little shady on showing these properties, so if someone could...

Ok, the proof looks simple since by defintion Cl A = intersection of all closed sets containing A. And textbooks give a quick proof that we all understand, but I have a question: Don't we first have to prove that a smallest closed set containing A exists in the first place? I'm trying to...

Let A\subset X be a subset of some topological space. If x\in\overline{A}\backslash A, does there exist a sequence x_n\in A so that x_n\to x? In fact I already believe, that such sequence does not exist in general, but I'm just making sure. Is there any standard counter examples? I haven't seen...

[SOLVED] metric space Homework Statement If x and y are two points in a metric space and d(x,y) = 1, is it always true that the closure of B(x,1/2) does not contain y? In general, is closure( B(x,r)) = \{z | r \geq d(x,z)\} Homework Equations The Attempt at a Solution

a. 1/n + 1/m : m and n are both in N b. x in irrational #s : x ≤ root 2 ∪ N c. the straight line L through 2points a and b in R^n. for part c. i got: intA= empty ; bdA=clA=accA=L Is this correct? how about part a and part b...i am so confused...

Homework Statement Suppose (X,d) is a metric space. For a point in X and a non empty set S (as a subset of X), define d(p,S) = inf({(d(p,x):x belongs to S}). Prove that the closure of S is equal to the set {p belongs to S : d(p,S) =0} Homework Equations Closure of S = S U S' , where S'...

Homework Statement X=R real numbers, U in T, the topology <=> U is a subset of R and for each s in U there is a t>s such that [s,t) is a subset of U where [s,t) = {x in R; s<=x<t} Find the closure of each of the subsets of X: (a,b), [a,b), (a,b], [a,b] The Attempt at a Solution I don't...

given a field F and two algebraic closures of F, are those two the isomorphic? and why doesn't this show that C and A (algebraic numbers) arent isomorphic?

Last night I had a bad dream :frown: In the dream I was sitting there with some of you fine people (I only remember Evo specifically) and we were all mourning the close-down of PF! Greg decided to change his career choice. It was a bad dream! :frown: :frown: ! It happened so fast too! We only...

I've started to read a book called "Types and Programming Languages" and this excerpt is from the section entitled "Mathematical Preliminaries". Now I can justify the conclusion to myself by reflecting that there is no tuple in R' that is neither a member of R nor (s,s) with s a member of...

Given a finite set G is closed under an associative product and that both cancellation laws hold in G,Then G must be a group. I need to prove that G must be a group, I understand that for this I only need to show that : 1) There exist the identity 2) There exist the inverse. But...

Hello, I am currently working on proving the following theorem The boundary \partial A and the closure \overline{A} of a subset A of \mathbb C are closed sets. Proof: Let A \subset \mathbb C. We want to show the set \partial A \cap \overline{A} is closed. To show that \partial A...

What is "notion of closure" May I know what is "notion of closure", I don't need comprehensive explanation, I just want to understand what is meant by a set is closed under another set in order to proceed. For example, Set B is closed under Set A, does it mean that Set A is a strict...

Define: \mathbb{C}((t)) = \{t^{-n_0}\sum_{i=0}^{\infty}a_it^i\ :\ n_0 \in \mathbb{N}, a_i \in \mathbb{C}\} What is its algebraic closure? My notes say that it is "close" to: \bigcup _{m \in \mathbb{N}}\mathbb{C}((t))(t^{1/m}) where \mathbb{C}((t))(t^{1/m}) is the extention of the...

Peter Woit called attention to the liklihood that new WMAP data will be coming out soon for some additional detail here is a post from Anthony Lewis (Cambridge) http://cosmocoffee.info/viewtopic.php?p=1391#1391He says that around 23 March there will be one or more conference talks about the...

Hi. Can somebody please check my work!? I'm just not sure about 2 things, and if they are wrong, all my work is wrong. 1. Find a counter example for "If S is closed, then cl (int S) = S I chose S = {2}. I am not sure if S = {2} is an closed set? I think it is becasue S ={2} does not have an...