Metric space Definition and 194 Threads

Homework Statement Let (X, d) be a metric space. The set Y in X , d(x; y) less than equal to r is called a closed set with radius r centred at point X. Show that a closed ball is a closed set. Homework Equations In a topological space, a set is closed if and only if it coincides...

Homework Statement Suppose A is an unbounded subspace of a metric space (X,d) (where d is the metric on X). Fix a point b in A let B(b,k)={a in X s.t d(b,a)<k where k>0 is a natural number}. Let A^B(b,k) denote the intersection of the subspace A with the set B(b,k). Then the...

Given a metric space (X,d), an element a \in X and a real number r>0, let A:= \{ x \in X | d(a,x) < r \}, C:= \{ x \in X | d(a,x) \leq r \} i need to show \bar{A} \subseteq C. The definition of the closure of A \subseteq X is \bar{A} = \cap_{C \subseteq X closed, A \subseteq C} C...

D={z in C | |z|<1} e: DxD -> R by e(z,w)=|(z-w)/(1-w'z)| (here the w'=the conjugate of w, not sure how to insert a bar on top of the w). Show that this is a metric space. It's all pretty easy till the triangle inequality (as always though, right?) so that's all I need to focus on. I'm...

Homework Statement The Wikipedia part of question 5 here: http://www.dpmms.cam.ac.uk/site2002/Teaching/IB/MetricTopologicalSpaces/2007-2008/Examples1.pdf Homework Equations All relevant information is given in the question above. The Attempt at a Solution I'm trying to simplify the...

Hello, why the set of all real numbers is complete metric space with euclidean metric? I know, that metric space is complete iff all sequences in it converges. But 1,2,3,4,... diverges. Thanx

I want to show the triangle inequality, d(x,x)=0, d(x,y)\neq0 for x\neqy implies that d(x,y)=d(y,x). Note that I do not have d(x,y)>0. But I know how to show this if I can get the transitive property. I have been trying to use the triangle ineq. to establish d(x,y)>=d(y,x) and...

Let X be a connected metric space, let a, b be distinct points of X and let r > 0. Is there a collection {B_i} of finitely many open balls of radius r such that their union is connected and contains a and b. I was trying to prove this by contradiction, but couldn't derive a contradiction. I...

Let (X,d) be a metric space A and B nonempty subsets of X and A is open. Show: A\capB = \oslash Iff A\capB(closure)= empty Only B closure it is easy to show rigth to left but how can i use A's open property I try to solve with contradiction s.t. there exist r>0 Br(p)\subseteqA\capB(closure) but...

Ptolemy metric space. Help! The problem is : "Let x,y,z,t belongs to R^n where d(x,y)=||x-y||. Show that(Ptolemy's inequality): d(x,y)d(z,t)<=d(x,z)d(y,t)+d(x,t)d(y,z)" I have found this related to the topic paper but I cannot show that the Euclidean space R^n is Ptolemy. The paper...

If I convert a symmetric group of degree n into a metric space, what metrics can be defined except a discrete metric? If a metric can be defined, I am wondering if the metric can describe some characteristics of a symmetric group.

Hello. Please, help me with this exercise: Let X be a topological space and let Y be a metric space. Let f_n: X \rightarrow Y be a sequence of continuos functions. Let x_n be a sequence of points of X converging to x. Show that if the sequence (f_n) converges uniformly to f then...

Homework Statement Let X be a non-empty set and let C be the set of all bounded real functions defined on X, with the metric induced by the supremum norm: d(f,g) = ||f - g|| = sup |f(x)-g(x)| , x in X. Show that the metric space (C,d) is complete. Hint: if \{f_{n}\} is a cauchy sequence...

Homework Statement We say that two metrics d, d' on a space S are equivalent if each "dominates" the other in the following sense: there exist constants M, M'>0 such that d'(x,y)<=M' d(x,y) and d(x,y)<=M d'(x,y) for all x,y in S. If metrics d, d' are equivalent, prove that (S,d) is...

Continuously differentiable Function C^1 {} \left[0,1\right] is complete with respect to the metric space D_\infty{}{f,g}=sup{\left|f(t)-g(t)\right|}+sup{\left|f^1{}(t)-g^1{}(t)\right|} but not in the d_\infty{}{f,g}=sup{\left|f(t)-g(t)\right|} Thanks for the helps in advance. Regards... BI

1. Any metric space can be converted into a topological space such that an open ball in a metric space corresponds to a basis in the corresponding topology (metric spaces as a specialization of topological spaces ). 2. Any topological space can be converted into a metric space only if there is a...

I don't know if this is more appropriate for the topology forum, but I am learning this in analysis. I am asked to say whether or not Q and N are homeomorphic to each other and to justify why. I am confused as to how to prove precisely that two spaces are homeomorphic, for there are no formal...

I just encountered a claim, that for any given metric space (X,d), there exists another topologically equivalent metric d' so that (X,d') is bounded. Anyone knowing anything about the proof for this?

Homework Statement Prove that every compact set is bounded. Homework Equations The usual compactness stuff - a compact set in a metric space X is one that, for every open cover, there is a finite subcover. The Attempt at a Solution I'm really hesitant about this question because my...

Homework Statement Let (X,d) be a metric space. Can a set E in X be both open and closed? Can a point in E be both isolated and an interior point? Homework Equations I've used the metric defined as d(x,y)=1 for x\ne y and 0 if x=y (we used this in a previous problem). I also used the...

Homework Statement Let X be an infinite set. For p,q \in X define: d(p,q) = {1 if p \neq q; 0 if p = q Suppose E is a finite subset of X, find all limit points of E. Homework Equations definition: a point p is a limit point of E if every neighborhood of p contains a point q \neq p...

[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

[SOLVED] Linear forms and complete metric space Homework Statement Question: Let L be a linear functional/form on a real Banach space X and let {x_k} be a sequence of vectors such that L(x_k) converges. Can I conclude that {x_k} has a limit in X? It would help me greatly in solving a certain...

1. Homework Statement Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi. 2. Homework Equations Hint from the book of Rudin: Center the point...

Homework Statement In R^2, define d(x,y)=smallest integer greater or equal to usual distance between x and y. Is d a metric for R^2? The Attempt at a Solution All is left is to show the triangle inequality is satisfied. Since the distances are rounded upwards I'd say yes.

I am having some troubles understanding the following, any help to me will be greatly appreciated. 1) Let S1 = {x E R^n | f(x)>0 or =0} Let S2 = {x E R^n | f(x)=0} Both sets S1 and S2 are "closed" >>>>>I understand why S1 is closed, but I don't get why S2 is closed, can anyone...

Can somebody give me an other metric space that is not dependent on the inner product i mean which is not derived from the inner product between two vectors.

Homework Statement Let M be an infinite metric space. Prove that M contains an open set U s.t. both U and its complement are infinite. Homework Equations The Attempt at a Solution For Euclidean spaces it is easy. You take (among other sets) R^{+} . However, I do not think that...

Homework Statement Show that if {K_n} is a decreasing family of compact connected sets in a metric space, then their intersection is connected as well. Illustrate with an example why 'compact' is necessary instead of just 'closed'. The Attempt at a Solution Well, I have a example for...

Homework Statement 'In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space.' http://en.wikipedia.org/wiki/Separable_metric_space Let...

This seems to be a very easy excercise, but I am completely stuck: Prove that in C([0,1]) with the metric \rho(f,g) = (\int_0^1|f(x)-g(x)|^2 dx)^{1/2} a subset A = \{f \in C([0,1]); \int_0^1 f(x) dx = 0\} is closed. I tried to show that the complement of A is open - it could be...

Is it by convention that all non trivial metric spaces have an infinite number of points? Just like all non trivial sequences has an infinite number of points.

Homework Statement Let X be an incomplete metric space. and Let X' denote its completion. I would like to show that there is Cauchy sequence in X which does not converge in X but does converge in X'. Moreover, I want to show that X contains every element of the sequence except the limit...

Homework Statement Can a complete metric space have empty interior? Homework Equations In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. The Attempt at a Solution But if M has no...

Homework Statement i am required to prove whether the following statement is true or false, Homework Equations there exists a metric space (X,d) with B1 contained in B2 contained in X such that B1=Bo(x1,3), B2=Bo(x2,2), and B2-B1 not equal to the empty set here Bo denotes the...

Homework Statement I am required to show that (C[0,1], || ||2) is a complete metric space, or to disprove that it is Homework Equations C[0,1] is the set of continuous functions on the bounded interval 0,1 The Attempt at a Solution I am immediately confused as I am told in my...

Homework Statement Defn: A subset A of a metric space (X, d) is NOWHERE DENSE if its closure has empty interior. Now I am told that this implies 1. A is nowhere dense iff closure of A does not contain any non-empty open set and 2. A is nowhere dense iff each non-empty open set has a...

I need to find a complete metric space and a sequence of nested, closed balls such that their infinite intersection is empty. How is this possible?

Hey guys, thanks for looking at this. Ok, so we're given the distance, d(x,C) between a point, x, and a closed set C in a metric space to be: inf{d(x,y): for all y in C}. Then we have to generalize this to define the distance between two sets I'm fairly certain you can define it as: the...

We recently discussed completion in my analysis class and I have a brief question on the subject. The completion X* of the metric space X is defined to be the set of Cauchy sequences of X with a defined equivalence relation ({xn}~{yn} if lim d(xn,yn)=0) and metric (D([xn],[yn])=lim d(xn,yn)). I...

Hi I have this here metric space problem which caused me some trouble: S \subseteq \mathbb{R}^n then the set \{ \| x - y \| \ | y \in S \} has the infimum f(x) = \{ \| x - y \| \ | y \in S \} where f is defined f: \mathbb{R}^n \rightarrow \mathbb{R} I have two problems here which I'm...

Hi I have another question in the field of analysis. Y \subseteq \mathbb{R}^n I'm suppose to show that if x \in \mathbb{R}^n, then the set \{ || x - y || \ y \in Y \} has an infimum, such that f(x) = \mathrm{inf} \{ || x - y || \ y \in Y \} I know that I'm suppose to show...

Let e>0 and let Be(r),Be(s) denote the open balls of radius e centered at r,s with respect to the p-adic metric. Prove that if r is an element of Be(s), then Be(s)=Be(r). Can someone show me how to use the strong triangle inequality to do this?

Here’s a problem I’ve been struggling with, for a while…. If (X,d) is a metric space and f:X-->X is a continuous function, then show that A={ x in X : f(x)=x} is a closed set. One possible way that I can think of is defining a new function g(x) = f(x)-x .Then A={x in X : g(x) =0}. Now {0} is...