# What is Metric space: Definition and 193 Discussions

In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

the distance from

A

{\displaystyle A}
to

B

{\displaystyle B}
is zero if and only if

A

{\displaystyle A}
and

B

{\displaystyle B}
are the same point,
the distance between two distinct points is positive,
the distance from

A

{\displaystyle A}
to

B

{\displaystyle B}
is the same as the distance from

B

{\displaystyle B}
to

A

{\displaystyle A}
, and
the distance from

A

{\displaystyle A}
to

B

{\displaystyle B}
is less than or equal to the distance from

A

{\displaystyle A}
to

B

{\displaystyle B}
via any third point

C

{\displaystyle C}
.A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.
The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. Some of non-geometric metric spaces include spaces of finite strings (finite sequences of symbols from a predefined alphabet) equipped with e.g. a Hamming's or Levenshtein distance, a space of subsets of any metric space equipped with Hausdorff distance, a space of real functions integrable on a unit interval with an integral metric

d
(
f
,
g
)
=

x
=
0

x
=
1

|

f
(
x
)

g
(
x
)

|

d
x

{\displaystyle d(f,g)=\int _{x=0}^{x=1}\left\vert f(x)-g(x)\right\vert \,dx}
or probabilistic spaces on any chosen metric space equipped with Wasserstein metric.

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1. ### 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...
2. ### 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...
3. ### 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...
4. ### 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...
5. ### 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...
6. ### 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...
7. ### 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))...
8. ### 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...
9. ### 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...
10. ### 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...
11. ### 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...
12. ### 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...
13. ### 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?
14. ### 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...
15. ### A Same open sets + same bounded sets => same Cauchy sequences?

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16. ### Complex numbers sequences/C is a metric space

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17. ### 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...
18. ### 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##...
19. ### 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...
20. ### I Baire Category Theorem: Question About Countable Dense Open Sets

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21. ### 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?
22. ### 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.
23. ### 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...
24. ### 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...
25. ### 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
26. ### 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...
27. ### 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...
28. ### 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...
29. ### 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...
30. ### 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
31. ### 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...
32. ### 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?
33. ### An empty ball in arbitrary metric space

Is it possible for a ball(with nonzero radius) to be empty in an arbitrary metric space?
34. ### 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...
35. ### 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.$$(-\infty,0)$$
36. ### 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...
37. ### 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.
38. ### 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}...
39. ### 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...
40. ### 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...
41. ### Sequence of metric spaces is compact iff each metric space is compact

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42. ### Limit of a sequence on a metric space

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43. ### Proving Uniformly Continuous Extension of Function ##f## in Metric Space ##E'##

Homework Statement Let ##S\subset E## where ##E## is a metric space with the property that each point of ##S^c## is a cluster point of ##S.## Let ##E'## be a complete metric space and ##f: S\to E'## a uniformly continuous function. Prove that ##f## can be extended to a continuous function...
44. ### Exploring the Metric Space Axioms: A Brief Overview and Exercise

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
45. ### Proving an open ball is connected in a metric space X

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...
46. ### Complete metric space can't have a countably infinite perfect space

1. Homework Statement . Let ##(X,d)## be a complete metric space. Prove that if ##P \subset X## is perfect, then P is not countably infinite. 3. The Attempt at a Solution . Well, I couldn't think of a direct proof, I thought that in this case it may be easier to assume is countably infinite...
47. ### Proving Closedness of a Set in a Metric Space

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...
48. ### A separable metric space and surjective, continuous function

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...
49. ### Is there a redshift in a conformally flat metric space?

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...
50. ### Complete countable metric space

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