Metric space Definition and 194 Threads

  1. E

    I No structure in ##(d,x)## in ##\textbf{Met*}(X)## is admissible

    The following question are taken from ##\textit{Arrows, Structures and Functors the categorical imperative}## by Arbib and Manes, and from ##\textit{Algebraic Theories}## by Manes ##\color{blue}{Question}\color{blue}{/difficulties:}## I am having a lot of difficulties doing the...
  2. O

    B Proper time of the observer resting in CMB reference frame

    Does t in a(t) in the FLRW metric correspond to the proper time of the immortal observer, who’s been resting in the CMB reference frame since its emission?
  3. cianfa72

    I Round 3-sphere symmetries as subspace of 4D Euclidean space

    As follow up of this thread in Special and General Relativity subforum, I'd like to better investigate the following topic. Consider the 4d euclidean space in which there are 10 ##\mathbb R##-linear independent KVFs. Their span at each point is 4 dimensional (i.e. at any point they span the...
  4. S

    I Do "bubble universes" in eternal inflation have their own spacetime?

    In the context of the model of eternal inflation, if an inflating "pocket universe" disconnects from an the background spacetime, does it mean that the baby universe itself can have its own spacetime? can they be described by a different spacetime metric than the background? if the original...
  5. P

    I On Borel sets of the extended reals

    On page 45 in Folland's text on real analysis, he writes that we define Borel sets in ##\overline{\mathbb R}## by ##\mathcal B_{\overline{\mathbb R}}=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}##. Then he remarks that this coincides with the usual definition of...
  6. P

    I Basic question on 'bounded implies totally bounded'

    Recall, a set ##X## is totally bounded if for each ##\epsilon>0##, there exists a finite number of open balls of radius ##\epsilon>0## that cover ##X##. Question: How can I verify that the balls ##B(\epsilon j,\epsilon)## cover ##T##? In particular, why the condition ##\epsilon |j_i|\leq 2b##...
  7. P

    Show inclusion map extends to an isometry

    I'm working an exercise on the completion of metric spaces. This exercise is from Gamelin and Greene's book and part of an exercise with several parts to it. I have already shown that ##\sim## is an equivalence relation, ##\rho## is a metric on ##\tilde X##, ##(\tilde X,\rho)## is complete and...
  8. P

    I Open balls dense in closed balls in Euclidean space

    Any set with at least two elements and equipped with the discrete metric is a counterexample to the claim that the closure of an open ball is a closed ball. Yet, in the back of the back book where they present solutions to some of their exercises, they write: I feel silly for asking, but I can...
  9. L

    Continuous functions on metric spaces part 2

    Hi, The task is as follows For the proof I wanted to use the boundedness, in the script of my professor the following is given, since both ##(X,d)## and ##\mathbb{R}## are normalized vector spaces I have now proceeded as follows ##|d(x,p)| \le C |x|## according to Archimedes' principle, a...
  10. S

    I Non-homogeneous and anisotropic metric and laws of physics...?

    In this popular science article [1], they say that if our universe resulted to be non-uniform (that is highly anisotropic and inhomogeneous) then the fundamental laws of physics could change from place to place in the entire universe. And according to this paper [2] anisotropy in spacetime could...
  11. chwala

    Understanding of the Metric Space axioms - (axiom 2 only)

    Am refreshing on Metric spaces been a while... Consider the axioms below; 1. ##d(x,y)≥0## ∀ ##x, y ∈ X## - distance between two points 2. ## d(x,y) =0## iff ##x=y##, ∀ ##x,y ∈ X## 3.##d(x,y)=d(y,x)## ∀##x, y ∈ X## - symmetry 3. ##d(x,y)≤d(x,z)+d(z,y)## ∀##x, y,z ∈ X## - triangle inequality...
  12. C

    I In Euclidian space, closed ball is equal to closure of open ball

    Problem: Let ## (X,d) ## be a metric space, denote as ## B(c,r) = \{ x \in X : d(c,x) < r \} ## the open ball at radius ## r>0 ## around ## c \in X ##, denote as ## \bar{B}(c, r) = \{ x \in X : d(c,x) \leq r \} ## the closed ball and for all ## A \subset X ## we'll denote as ## cl(A) ## the...
  13. K

    I Is this generalization equivalent to usual Aposyndetic

    For some basic definitions we call connected, metric space a continuum and we say that continuum is aposyndetic if for every pair of points p,q exists a subcontinuum W such that $p \in int(W) \subset W \subset X \setminus \{q\}$ similarly I introduce a notion of "zero set aposyndetic" as: X is...
  14. facenian

    I A problem of completeness of a metric space

    Hi, I found this problem in Munkres' topology book, and it seems to be contradictory: Let X be a metric space. (a) Suppose that for some ϵ>0, every ϵ-Ball in X has compact closure. Show that X is complete. (b) Suppose that for each x\in X there is an \epsilon>0 such as the ball B(x,\epsilon) has...
  15. Norashii

    Proof of Subspace Topology Problem: Error Identification & Explanation

    I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance.**My proof:** Take a limit point x of U that is not in U, but is in K (in other words x \in K \cap(\overline{U}-U))...
  16. Math Amateur

    MHB Open Sets in a Discrete Metric Space .... ....

    In a discrete metric space open balls are either singleton sets or the whole space ... Is the situation the same for open sets or can there be sets of two, three ... elements ... ? If there can be two, three ... elements ... how would we prove that they exist ... ? Essentially, given the...
  17. Math Amateur

    MHB The Metric Space R^n and Sequences .... Remark by Carothers, page 47 ....

    I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms ... ... I need help with a remark by Carothers concerning convergent sequences in \mathbb{R}^n ...Now ... on page 47 Carothers writes the following: In the above text from Carothers we...
  18. Math Amateur

    MHB Understand Example 3.10 (b) Karl R. Stromberg, Chapter 3: Limits & Continuity

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows: My question is as...
  19. Math Amateur

    MHB Countably Dense Subsets in a Metric Space .... Stromberg, Lemma 3.44 .... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows: In the above...
  20. Math Amateur

    MHB Open Subsets in a Metric Space .... Stromberg, Theorem 3.6 ... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows: In the above...
  21. CCMarie

    B Finite distance between two points

    If X is a metric space, what is the simplest sufficient condition for d(x,y)<∞, ∀ x, y ∈ X?
  22. L

    A Structure preserved by strong equivalence of metrics?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...
  23. L

    A Same open sets + same bounded sets => same Cauchy sequences?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
  24. F

    Complex numbers sequences/C is a metric space

    Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...
  25. T

    Metric space of continuous & bounded functions is complete?

    Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
  26. T

    I Proof Explanation: Showing an extension to a continuous function

    I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...
  27. T

    Is This Approach Valid for Proving the Discrete Metric in a Metric Space?

    Homework Statement Let ##x,y\in X## such that ##X## is a metric space. Let ##d(x,y)=0## if and only if ##x=y## and ##d(x,y)=1## if and only if ##x\neq y## Homework Equations N/A The Attempt at a Solution I have already seen various approaches in proving this. Although, I just want to know if...
  28. mr.tea

    I Baire Category Theorem: Question About Countable Dense Open Sets

    Hi, I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space. Assume that we have the countable collection of dense open sets ##...
  29. Zafa Pi

    I Chaos like phenomena on a simple metric space?

    Let M = {p, x1, x2, x3, ...} be a metric space with no isolated points. f: M → M is continuous with f(xn) = xn+1, and f(p) = p. We say f separates if ∃ δ > 0, ∋ for any y and z there is some n with |fn(y) - fn(z)| > δ, where fn+1(y) = f(fn(y)). QUESTION: Does f separate?
  30. facenian

    I Compact subspace in metric space

    Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.
  31. R

    I Understanding metric space definition through concrete examples

    Right now, I am studying Advanced Calculus (proof based), and it is hard thinking through some of the definitions without first thinking about it concretely (meaning how to visualize it better geometrically, if that makes any sense?). I need help with this definition. Definition Let X be a...
  32. C

    I Where the scale factor a(t) appears in the metric

    Hello, I was enjoying Zee's book on GR when I noticed the location of this "a(t)" thing in the metric sound quite disturbing to me. BTW: I experience the same annoyance and went down to the same conclusions, when I watched a related Theoretical Minimum lesson...Here's the setup, the flat...
  33. Lucas SV

    I Metrics which generate topologies

    Given a topological space ##(\chi, \tau)##, do mathematicians study the set of all metric functions ##d: \chi\times\chi \rightarrow [0,\infty)## that generate the topology ##\tau##? Maybe they would endow this set with additional structure too. Are there resources on this? Thanks
  34. I

    Is a Complete Subspace Necessarily Closed in a Metric Space?

    Homework Statement Let ##E## be a metric subspace to ##M##. Show that ##E## is closed in ##M## if ##E## is complete. Show the converse if ##M## is complete. Homework Equations A set ##E## is closed if every limit point is part of ##E##. We denote the set of all limit points ##E'##. A point...
  35. I

    Convergence of sequence in metric space proof

    Homework Statement Let ##E \subseteq M##, where ##M## is a metric space. Show that ##p\in \overline E = E\cup E' \Longleftrightarrow## there exists a sequence ##(p_n)## in ##E## that converges to ##p##. ##E'## is the set of limit points to ##E## and hence ##\overline E## is the closure of...
  36. L

    How to prove the following defined metric space is separable

    Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to...
  37. Samuel Williams

    Does T have a unique fixed point in X?

    Let (X, d) be a complete metric space, and suppose T : X → X is a function such that T^2 is a contraction. [By T^2, we mean the function T^2 : X → X given by T^2(x) = T(T(x))]. Show that T has a unique fixed point in X. So I have an answer, but I am not sure whether it is correct. It goes as...
  38. lucasLima

    Help proving triangle inequality for metric spaces

    So, i need to proof the triangle inequality ( d(x,y)<=d(x,z)+d(z,y) ) for the distance below But I'm stuck at In those fractions i need Xk-Zk and Zk-Yk in the denominators, not Xk-Yk and Xk-Yk. Thanks in advance
  39. Rasalhague

    Proving that Every Closed Set in Separable Metric Space is Union of Perfect and Countable Set

    Homework Statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.) Homework Equations Every separable metric space has a countable base. The...
  40. X

    Homemorphism of two metric space

    If I create a bijective map between the open balls of two metric spaces, does that automatically imply that this map is a homemorphism?
  41. X

    An empty ball in arbitrary metric space

    Is it possible for a ball(with nonzero radius) to be empty in an arbitrary metric space?
  42. C

    Showing a metric space is complete

    Homework Statement Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)| is a complete metric space. The Attempt at a Solution Spent a few hours just thinking about this question, trying to prove...
  43. S

    MHB Complete Metric Space: X, d | Analysis/Explanation

    Hi i am confused of the following question. Suppose we have a Metric Space (X,d), where d is the usual metric. Now are the following subsets complete, if so why?? 1.$$X=[0,1]$$ 2.$$X=[0,1)$$ 3.$$X=[0,\infty)$$ 4.$$(-\infty,0)$$
  44. phosgene

    Show that T is a contraction on a metric space

    Homework Statement Consider the metric space (R^{n}, d_{∞}), where if \underline{x}=(x_{1}, x_{2}, x_{3},...,x_{n}) and \underline{y}=(y_{1}, y_{2}, y_{3},...,y_{n}) we define d_{∞}(\underline{x},\underline{y}) = max_{i=1,2,3...,n} |x_{i} - y_{i}| Assume that (R^{n}, d_{∞}) is...
  45. E

    MHB About open sets in a metric space.

    Let $$(E=]-1,0]\cup\left\{1\right\},d) $$ metric space with $$d$$ metric given by $$d(x,y)=|x-y|$$, and $$||$$absolute value. How I can find open sets of E explicitly? Thanks in advance.
  46. T

    What Defines a Ball, Interior, and Limit Point in Metric Spaces?

    Homework Statement For a metric space (X,d) and a subset E of X, de fine each of the terms: (i) the ball B(p,r), where pεX and r > 0 (ii) p is an interior point of E (iii) p is a limit point of E Homework Equations The Attempt at a Solution i) Br(p) = {xεX: d(x.p)≤r}...
  47. B

    Need help with simple proof, metric space, open covering.

    Please take a look at the proof I added, there are some things I do not understand with this proof. 1. Does it really show that |f(x)-f(y)|≤d(x,y) for all x and y? Or does it only show that if there is an ball with radius r around x, and this ball is contained in an O in the open covering, and...
  48. M

    Statement about topology of subsets of a metric space.

    Homework Statement . Prove that a closed subset in a metric space ##(X,d)## is the boundary of an open subset if and only if it has empty interior. The attempt at a solution. I got stuck in both implications: ##\implies## Suppose ##F## is a closed subspace with ##F=\partial S## for some...
  49. M

    Sequence of metric spaces is compact iff each metric space is compact

    Homework Statement . Let ##(X_n,d_n)_{n \in \mathbb N}## be a sequence of metric spaces. Consider the product space ##X=\prod_{n \in \mathbb N} X_n## with the distance ##d((x_n)_{n \in \mathbb N},(y_n)_{n \in \mathbb N})=\sum_{n \in \mathbb N} \dfrac{d_n(x_n,y_n)}{n^2[1+d_n(x_n,y_n)]}##...
  50. M

    Limit of a sequence on a metric space

    Homework Statement . Let ##(X,d)## be a metric space and let ##D \subset X## a dense subset of ##X##. Suppose that given ##\{x_n\}_{n \in \mathbb N} \subset X## there is ##x \in X## such that ##\lim_{n \to \infty}d(x_n,s)=d(x,s)## for every ##s \in D##. Prove that ##\lim_{n \to \infty} x_n=x##...
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