What is closure: Definition and 169 Discussions

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.

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 Suppose R is a relation on S, and define R' = R \cup \{(s,s) \vert s \in S. Prove that R' is the reflexive closure. Homework Equations The reflexive closure of R is the smallest reflexive relation R' that contains R. That is, if there is another R'' that contains R...

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...

Homework Statement F = {S -> NOPQZVWX, NX -> UV, NXT -> RY, YQ -> NXR, QZ -> S, PQ -> S} Find the minimum closure of F. Homework Equations none The Attempt at a Solution So far I have made RHS of each FD into a single attribute and then I eliminated redundant attributes from...