Metric space Definition and 198 Threads

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...

Hi there, I came across the following problem and I hope somebody can help me: I have some complete metric space (X,d) (non-compact) and its product with the reals (R\times X, D) with the metric D just being D((t,x),(s,y))=|s-t|+d(x,y) for x,y\in X; s,t\in R. Then I have some sequences...

Hi, All: Let X be a metric space and let A be a compact subset of X, B a closed subset of X. I am trying to show this implies that d(A,B)=0. Please critique my proof: First, we define d(A,B) as inf{d(a,b): a in A, b in B}. We then show that compactness of A forces the existence of a in A...

Homework Statement Let X and Y be subsets of R^2, both non-empty. If X is open, the the sum X+Y is open. This is either supposed to be proved or disproved. Homework Equations The Attempt at a Solution This strikes me as false since we are only given the X is open...

Homework Statement Let A be a non-empty set and let d be the discrete metric on X. Describe what the open subsets of X, wrt distance look like. Homework Equations The Attempt at a Solution I think that the closed sets are the subsets of A that are the complement of a union of...

Homework Statement show that (the range of) a sequence of points in a metric space is in general not a closed set. Show that it may be a closed set. 2. The attempt at a solution I don't know where to start. For example, if we are given a sequence of real numbers and the distance...

Can someone help me understand the notion of complete metric space? I've read the definition (the one involving metrics that go to 0), but I can't really picture what it is. Does anyone have any examples that could help me understand this?

Homework Statement Prove that the sequence space l^2 (the set of all square-summable sequences) is complete in the usual l^2 distance. Homework Equations No equations.. just the definition of completeness and l^2. The Attempt at a Solution I have a sample proof from class to show...

Homework Statement Let X be the integers with metric p(m,n)=1, except that p(n,n)=0. Show X is closed and bounded but not compact. Homework Equations I already check the metric requirement. The Attempt at a Solution I still haven't got any clue yet. Can anyone help me out?

Homework Statement Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0. Homework Equations The Attempt at a Solution First, let f:(M,d)->(N,p) be uniformly continuous...

Homework Statement X, Y are metric spaces and f: X \rightarrow Y Prove that f is discontinuous at a point x \in X if and only if there is a positive integer n such that diam f(G) \geq 1/n for every open set G that contains x Homework Equations diameter of a set = sup{d(x,y): x,y...

Homework Statement Let (X, d) be a metric space and let A,B\subsetX be two disjoint closed sets. Show that X is normal by using the function f(x)=d(x,A)/[d(X,A)+d(x,B)] The Attempt at a Solution I'm somewhat stuck on this. I'm guessing the proof is pretty short, but any help would be...

Homework Statement Prove that if a metric space (X, d) has an \epsilon-net for some positive number \epsilon, then (X, d) is bounded. Homework Equations The Attempt at a Solution I think that (X, d) might be not bounded. For example, let X be a subspace of real line with usual...

Homework Statement Suppose M is a metric space and A \subseteq M. Then A is totally bounded if and only if, for every \epsilon >0, there is a finite \epsilon-dense subset of A. Homework Equations The Attempt at a Solution I have already done the \Rightarrow but need to verify...

Is component(maximal connected set) of a metric space open or closed or both(clospen)?or even can be half-open(not open and not closed)? I know it is a silly question as (3,5] is a component in R,right? However some theorem i encountered stated that component must be closed or must be open. I...

Homework Statement One needs to show that a connected metric space having more than one point is uncountable. The Attempt at a Solution First of all, if (X, d) is a connected metric space, it can't be finite, so assume it's countably infinite. Let x be a fixed point in X. For any x1 in...

Hello all,Here is my question while reading a proof. For a compact set K in a separable metrizable spce (E,\rho) and a continuous function t \mapsto f(t) , if we define D_{K} = \inf \{ t \geq 0 \; : \; f(t) \in K \} then, D_{K} \leq t if and only if \inf\{ \rho(f(q),K) : q \in...

Any hint PLZ [PLAIN]http://img151.imageshack.us/img151/1715/1111111111k.jpg Thank You

is minkowski space a metric space. As best as i can remember a metric space is a set with a metric that defines the open sets. With this intuition is Minkowski space a metric space. I mean i think it should be, but according to one of the requirements for a metric: d(x,y)=0 iff x=y triangle...

Metric given by (d) in the textbook on page 48 in the following url. How could we plot this function and characterize what the metric looks like for varying a, b, c. Start by plotting the case when d(x,0)=1, a=b=c=1. Vary a, b, and c individually. Show plots for each case and Characterize how...

Homework Statement Consider the sequence a_1,a_2,..., such that \lim_{n\rightarrow\infty} a_n = a (with a_i \in R). Show that \lim_{n\rightarrow\infty}\left(\frac{\sum_{i=1}^n a_i}{n}\right) = a In other words, it's given that for some \epsilon > 0,d(a_n,a) < \epsilon\ \forall n > N...

ok i am stumped on a proof i am trying to construct of a metric: d(x,y)=\frac{|x-y|}{1+|x-y|} so, out of the 3 requirements to be a metric, the first 2 are trivial and I am just working on proving the triangle inequality... i need \frac{|x-y|}{1+|x-y|} \leq \frac{|x-z|}{1+|x-z|} +...

Homework Statement prove that the given function is a metric, or give a counterexample to show how it fails to be a metric: d(x,y)=|x3-y3| Homework Equations ok, out of the 3 requirements to be a metric, 2 are trivial. The third is to prove the triangle inequality holds...

Homework Statement Suppose that (x_n) is a sequence in a compact metric space with the property that every convergent subsequence has the same limit x. Prove that x_n \to x as n\to \infty Homework Equations Not sure, most of the relevant issues pertain to the definitions of the space...

Homework Statement To prove that every metric space is regular. :) The Attempt at a Solution So, a regular space satisfies the T1 and T3-axioms. For T1: Let a, b be two distinct points of a metric space (X, d). Then d(a, b) > 0, and let r = d(a, b)/2. Then the open ball K(a, r) is a...

As the title suggests, I have to prove, if (X, d) is a metric space, that for any subset A of X, diam A = diam Cl(A), i.e. the diameter of A equals the diameter of its closure. So, if A is closed, it is trivial, since Cl(A) = A. Assume A is open. Now I'm a bit lost. If A is open in (X, d)...

Hello my friends! My textbook has the following statement in one of its chapters: Chapter 8:Topology of R^n If you want a more abstract introduction to the topology of Euclidean spaces, skip the rest of this chapter and the next, and begin Chapter 10 now. Chapter 10 covers topological...

So, I have to prove that in the metric space (R^n, d), where d is the standard Euclidean metric, B(x1, r1) = B(x2, r2) <==> x1 = x2 & r1 = r2. I finished the proof, but I'm not sure about one step. Assume B(x1, r1) = B(x2, r2) with x1 = x2. Using the triangle inequality for x1, x and x2...

Homework Statement A metric on C[0,1] is defined by: d(f,g) = ( \int_0^1 \! (f(x) - g_t(x))^2 \, dx )^{1/2} Find t e R such that the distance between the functions f(x) = e^x - 1 and g_t(x) = t * x is minimal. Homework Equations Given above The Attempt at a Solution The first...

Homework Statement Let (X,\theta) be a metric space. Take K > 0 and define. \theta : X \cross X \rightarrow \real_{0}^{+}, (x,y)\rightarrow \frac{K\phi(x,y)}{1+K\phi(x,y)} Show that (X,\theta) is a metric space. Homework Equations can someone please check my triangle...

Homework Statement I am using Rosenlicht's Intro to Analysis to self-study. 1.) I learn that the complements of an open ball is a closed ball. And... 2.) Some subsets of metric space are neither open nor closed. Homework Equations Is something amiss here? I do not understand how...

EDIT: I figured out my error, so don't worry about reading through all of this unless you find it an interesting problem Homework Statement This is Baby Rudin's exercise 2.27: http://img63.imageshack.us/img63/584/fool.png Instead of proving for R^k, I did it for an arbitrary separable...

Homework Statement Let (X,d) be a metric space, and let A,B \subset X be disjoint closed subsets. 1. Construct a continuous function f : X \to [0,1] such that A \subseteq f^{-1}({0}) and B \subseteq f^{-1}({1}). Hint: use the functions below. 2. Prove that there are disjoint sets U,V...

Let (X,d) be a metric space. Show that if there exists a metric d' on X/~ such that d(x,y) = d'([x],[y]) for all x,y in X then ~ is the identity equivalence relation, with x~y if and only if x=y. i have: assume x=y then d(x,y)=0 and [x]=[y] which implies d'([x],[y])=0 also. now...

Homework Statement Let X = \mathbb{R}^n be equipped with the metric d_p(\boldsymbol{x}, \boldsymbol{y}) := \left[ \sum^n_{i=1} |x_i - y_i|^p \right]^{\frac{1}{p}}, p \geq 1 Homework Equations Show that if p < 1 then d_p is not a metric. The Attempt at a Solution I don't know what...

Homework Statement the problem: Let M be a metric in which the closure of every open set is open. Prove that M is discrete The Attempt at a Solution To show M is discrete, it's enough to show every singleton set in M is open. For any x in M, assume it's not open, then there exist a...

Homework Statement Prove that if (X,\rho) is a metric space then so is (X,\bar\rho), where \bar\rho:X \times X \Rightarrow R_{0}^{+}, (x,y) \Rightarrow \frac{\rho(x,y)}{1+\rho(x,y)}. Homework Equations I'm trying to prove the axiom that a metric space is positive definate...

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...

For a metric space (X,d), prove that a Cauchy sequence {xn} has the property d(xn-xn+1)--->0 as n--->\infty In working this proof, is it really as simple as letting m=n+1?

Can we have some examples in which a nowhere dense subset of a metric space is not closed?

"Closed" set in a metric space Homework Statement 1) Let (X,d) be a metric space. Prove that a "closed" ball {x E X: d(x,a) ≤ r} is a closed set. [SOLVED] 2) Suppose that (xn) is a sequence in a metric space X such that lim xn = a exists. Prove that {xn: n E N} U {a} is a closed subset of...

Let (X,d) be a metric space. d is a metric. 1) Is it possible that d(1,2)=d(1,8)? 2) Is it possible that d(1,3)>d(1,100)? If the answer is yes, wouldn't it be weird? The distance between 1 and 3 is larger than the distance between 1 and 100? This is highly counter-intuitive to me... 3)...

Homework Statement let (X,\sigma) be a metric space. xyz \in Rshow that \mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y) Homework Equations The Attempt at a Solution \mid \sigma(x,z)-\sigma(y,z) \mid \leq \sigma(x,y)=\mid \sigma(x,y) \mid = \mid\sigma (z,x) +...

Homework Statement Homework Equations N/A The Attempt at a Solution I'm really not having much progress on this question. My thoughts are as shown above.

Homework Statement Give an example of a decreasing sequence of closed balls in a complete metric space with empty intersection. Hint 1: use a metric on N topologically equivalent to the discrete metric so that {n≥k} are closed balls. In={n,n+1,n+2,...}. Homework Equations N/A The...

Let (X,d) be a metric space, and x is an element in X. Show that \{y \in X|d(y,x)>r\} is open for all r in Reals. I really need some help with this one, I have almost no idea on how I am meant to solve this. The only thing i know is that I have to use the Openness definition, that states...

1) Fact: Let X be a metric space. Then the set X is open in X. Also, the empty set is open in X. Why?? 2) Let E={(x,y): x>0 and 0<y<1/x}. By writing E as a intersection of sets, and using the following theorem, prove that E is open. Theorem: Let X,Y be metric spaces. If f:X->Y is...

Homework Statement Give an example of a decreasing sequence of closed balls in a COMPLETE metric space with empty intersection. Hint 1: use a metric on N topologically equivalent to the discrete metric so that {n≥k} are closed balls. Hint 2: In={n,n+1,n+2,...}. Consider the metric...

Hi I have two questions, 1. A metric space is an ordered pair (M,d) where M is a set (which some authors require to be non-empty) and d is a metric on M, that is, a function d : M x M -> R ------------From Wikipedia. http://en.wikipedia.org/wiki/Metric_space#Definition I just...