Metric space Definition and 194 Threads

Quick question about the metric space axioms, is the requirement that the distance function be positive-semidefinite an axiom for metric spaces? It seems that it can be proved from the other axioms (symmetry, identity of indiscernibles and the triangle inequality). BiP

Homework Statement . Let ##B(a,ε) (ε>0)## in a metric space ##(X,d)##. Decide whether this subset of ##(X,d)## is connected or not. The attempt at a solution. Well, I know open intervals in the real line are connected. I suppose that an open ball in a given metric space can be imagined...

Homework Statement Prove that if lim n→∞ (p_n ) = p in a metric space then the set of points {p,p_1,p_2, ...,} are closed. 2. Relevant information The definition of close in my book is "a set is closed if and only if its complementary is open." So I want to prove this by contradiction. I...

Homework Statement . Let X, Y be metric spaces and ##f:X→Y## a continuous and surjective function. Prove that if X is separable then Y is separable. The attempt at a solution. I've tried to show separabilty of Y by exhibiting explicitly a dense enumerable subset of Y: X is separable...

Hello PF: I noticed a thread on PF in which TOM STOER and others were discussing how to calculate the redshift for an arbitrary metric. I need to talk to Tom if he is still on this list. The question has arisen in an applied physics field whether the following conformally flat metric...

Homework Statement It is clear that a countable complete metric space must have an isolated point,moreover,the set of isolated points is dense.what example is there of a countable complete metric space with points that are not isolated? Homework Equations The Attempt at a Solution

I must be overlooking something! Given a metric space (E,d), the improper subset E is open in E. How? Here is my understanding: 1) We call a set S(subset of E) open iff for all x(element of S) there exist epsilon such that an open ball of radi epsilon centered about s is wholly contained in...

Homework Statement A dissimilarity measure d(x, y) for two data points x and y typically satisfy the following three properties: 1. d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y 2. d(x, y) = d(y, x) 3. d(x,z) ≤ d(x, y) + d(y,z)The following method has been proposed for measuring the...

Hi everyone, I posted this a couple days ago and didn't get a response, so I thought I'd try again. Let me know if something about this is confusing. Thanks! Homework Statement Let X be a metric space and let x\in{X} be any point. Prove that the set \left\{x\right\} is closed in X...

Metric Space and Topology HW help! Let X be a metric space and let (sn )n be a sequence whose terms are in X. We say that (sn )n converges to s \ni X if \forall \epsilon > 0 \exists N \forall n ≥ N : d(sn,s) < \epsilon For n ≥ 1, let jn = 2[(5^(n) - 5^(n-1))/4]. (Convince yourself...

Homework Statement Let X be an infinite set. For p\in X and q\in X, d(p,q)=1 for p\neq q and d(p,q)=0 for p=q Prove that this is a metric. Find all open subsets of X with this metric. Find all closed subsets of X with this metric. Homework Equations NA The Attempt at a...

Homework Statement Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M. Homework Equations Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists...

If every continuous function on M is bounded, what does this mean? I am not sure what this function actually is... is it a mapping from M -> M or some other mapping? Is the image of the function in M? Any help would be greatly appreciated!

Hi, If X \LARGE is a metric space and E \subset X is a discrete set then is E \LARGE open or closed or both? Here's my understanding: E \LARGE is closed relative to X \LARGE. proof: If p \subset E then by definition p \LARGE is an isolated point of E \LARGE, which implies that p...

Hi, In Baby Rudin, Thm 3.6 states that If p(n) is a sequence in a compact metric space X, then some subsequence of p(n) converges to a point in X. Why is it not the case that every subsequence of p(n) converges to a point in X? I would think a compact set would contain every sequence...

Homework Statement show the set {f: ∫f(t)dt>1(integration from 0 to 1) } is an open set in the metric space ( C[0,1],||.||∞） and if A is the subset of C[0,1] defined by A={f:0<=f<=1} is closed in the norm ||.||∞ norm. Homework Equations C[0,1] is f is continuous from 0 to 1.and ||.||∞...

The usual proof of this theorem seems to assume that the topology of the metric space is the one generated by the metric. But if I use another topology, for example the trivial, the space need not be Hausdorff but the metric stays the same. Am I missing something or is the statement of the...

Like any mathematical relativity between them as per General Relativity?

http://imageshack.us/a/img12/8381/37753570.jpg I am having trouble with this question, like I do with most analysis questions haha. It seems like I must show that the maximum exists. So E is compact -> E is closed To me having E closed seems like it is clear that a maximum distance...

I'm beginning self-study of real analysis based on 'Introductory Real Analysis' by Kolmogorov and Fomin. This is from section 5.2: 'Continuous mappings and homeomorphisms. Isometric Spaces', on page 45, Problem 1. This is my first post to these forums, but I'll try to get the latex right...

I have a simple question about the notation. I want to be more correct with notation, I don't understand exactly what the notation is saying. In regards to a Metric space A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function {{\bf{d: M \times...

Homework Statement Show that the following three conditions of a metric space imply that d(x, y)=d(y, x): (1) d(x, y)>=0 for all x, y in R (2) d(x, y)=0 iff x=y (3) d(x, y)=<d(x, z)+d(z, y) for all x, y, z in R (Essentially, we can deduce a reduced-form definition of a metric space...

Homework Statement let d be a metric on an infinite set m. Prove that there is an open set u in m such that both u amd its complements are infinite. Homework Equations If d is not a discrete metric, and M is an infinite set (uncountble), then we can always form an denumerable subset...

Hi all, Given a metric space (X,d), one can take its completion by doing the following: 1) Take all Cauchy sequences of (X,d) 2) Define a pseudo-metric on these sequences by defining the distance between two sequences to be the limit of the termwise distance of the terms 3) Make this a...

Homework Statement Let X be a metric space and A a subset of X. Prove that the following are equivalent: i. A is dense in X ii. The only closed set containing A is X iii. The only open set disjoint from A is the empty set Homework Equations N/A The Attempt at a Solution I can...

Suppose that (X,d) is a metric Show \tilde{d}(x,y) = \frac{d(x,y)}{\sqrt{1+d(x,y)}} is also a metric I've proven the positivity and symmetry of it. Left to prove something like this Given a\leqb+c Show \frac{a}{\sqrt{1+a}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} I try to...

Hello, the following is a post that was in progress and I am continuing it here after I received a message saying that most of the members had moved from mathhelpforum here. Me: I have a problem where I am asked to show that for a complete metric space X, the the natural Isometry F:X --> X* is...

(1) (X,d) is a COMPACT metric space. (2) f:X->X is a function such that d(f(x),f(y))=d(x,y) for all x and y in (X,d) Prove f is onto. Things I know: (2) => f is one-one. (2) => f is uniformly continuous. I tried to proceed by assuming the existence of y in X such that y has no...

Homework Statement Is empty set a metric space? Homework Equations None. The Attempt at a Solution It seems so because all the metric properties are vacuously satisfied. Mabe the question had better be put like this: Does mathematicians tend to think empty set as a metric space...

A metric space of equivalent Cauchy sequence classes (Z, rho) is defined using a metric of the sequence elements in the space (X,d), where d is from XX to R (real numbers). The metric of the sequence classes is rho = lim d(S, T), where S and T are the elements of the respective sequences. To...

Homework Statement Consider a metric space (X,d) with subsets A and B of X, where A and B have non-zero intersection. Show that diam(A\bigcupB) \leq diam(A) + diam(B) Homework Equations The Attempt at a Solution A hint would be very much appreciated. :smile:Let x\inA, y\inB, z\inA\bigcupB...

Homework Statement Let f be a continuous real function on a metric space X. Let Z(f) be the set of all p in X at which f(p) = 0. Prove that Z(f) is closed. Homework Equations Definition of continuity on a metric space. The Attempt at a Solution Proof. We'll show that X/Z(f) = {p...

Homework Statement For x,y \in\mathbb{R} define a metric on \mathbb{R} by d_2(x,y) = |\tan^{-1}(x) - \tan^{-1}(y) | where \tan^{-1} is the principal branch of the inverse tangent, i.e. \tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2). If (x_n)_{n\in\mathbb{N}} is a sequence in \mathbb{R} and...

Homework Statement The Attempt at a Solution I've got through this question up to the last bit. I've got B(0,1) = \{0\} and B(0,2) = \{y\in\mathbb{R} : -1<y<1 \} (i.e. the open interval (-1,1).) How do I show that every subset of \mathbb{R} is open (A \subseteq X is open if it...

Hi! I'm a beginner for a subject "topology". While studying it, I found a confusing concept. It makes me crazy.. I try to explain about it to you. For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function. I've also learned that for a set...

If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?

Hello. In my analysis book, it says that "Any closed bounded subset of E^n is compact" where E is an arbitrary metric space. I looked over the proof and it used that fact that E^n was complete, but it does not say that in the original condition so I was wondering if the book made a mistake in...

Can we define a metric space (\emptyset, d)? The metric is the part that confuses me, since it seems like all of the required properties of d are satisfied since they are "not not satisfied", but I'm not sure. Thank you!

Homework Statement Assume some metric space (K,d) obeys Lindelof, take (X,d) a metric subspace of (K,d) and show it too must obey Lindelof. The Attempt at a Solution I'm assuming since I know that (K,d) obeys Lindelof then there is some open cover that has a countable subcover say {Ji | i is a...

Homework Statement Regard Q, the set of all rational numbers, as a metric space, with d(p, q) = |p − q|. Let E be the set of all p ∈ Q such that 2 < p2 < 3. Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q? Homework Equations Definition of interior...

Homework Statement Let X and Y be metric spaces, f a function from X to Y: a) If X is a union of open sets Ui on each of which f is continuous prove that f is continuous on X. b) If X is a finite union of closed sets F1, F2, ... , Fn on each of which f is continuous, prove that f is continuous...

" Let X be a metric space and p be a point in X and be a positive real number. Use the definition of openness to show that the set U(subset of X) given by: U = {x∈X|d(x,p)>r} is open. " I have tried: U is open if every point of U be an interior point of U. x is an interior point of U if there...

Hey All, I have been working on some Metric Space problems for roughly 20hrs now and I cannot seem to grasp some of these concepts so I was hoping someone here could clear a few things up for me. My first problem is detailed below... I have the following metric... d(x,y) = d(x,y)/(1 +...

Homework Statement Prove that \rho_{0}(x,y)=max_{1 \leq k \leq n}|x_{k} - y_{k}|=lim_{p\rightarrow\infty}(\sum^{n}_{k=1}|x_{k}-y_{k}|^{p})^{\frac{1}{p}} Homework Equations The Attempt at a Solution My approach was to define a_{m}=max_{1 \leq k \leq n}|x_{k} - y_{k}| and...

Homework Statement Given (R+, d), R-Real # d= | ln(x/y) | Show that this metric space is complete Homework Equations The Attempt at a Solution Firstly, I know that to show it is complete I need to have that all Cauchy sequences in that space converge... So I'm not 100%...

Hello, I was wondering if if has any sense of talking about angles on an arbitrary http://en.wikipedia.org/wiki/Metric_space" (where only a distance function with some properties is defined) At first sight it seems to not has any sense, only some metric spaces has angles, namely does that...

I'm trying to verify that if (M,d) is a metric space, then (N,e) is a metric space where e(a,b) = d(a,b) / (1 + d(a,b)). Everything was easy to verify except the triangle inequality. All I need is to show that: a <= b + c implies a / (1 + a) <= (b / (1 + b)) + (c / (1 + c) Any help...

Suppose (X,d) is a metric space and A, a subset of X, is closed and nonempty. For x in X, define d(x,A) = infa in A{d(x,a)} Show that d(x,A) < infinity. I really don't have much of an idea on how to show it must be finite. An obvious thought comes to mind, namely that a metric is...

I've encountered the term Hausdorff space in an introductory book about Topology. I was thinking how a topological space can be non-Hausdorff because I believe every metric space must be Hausdorff and metric spaces are the only topological spaces that I'm familiar with. my argument is, take two...

Homework Statement Having a little trouble on number 24 of Chapter 3 in Rudin's Principles of Mathematical Analysis. How do I prove that the completion of a metric space is complete? Homework Equations X is the original metric space, X^* is the completion, or the set of...