What is Metric space: Definition and 192 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.
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...
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...
Completion of Metric Space Proof from "Intro. to Func. Analysis w/ Applications"
Homework Statement
I have started studying Functional Analysis following "Introduction to Functional Analysis with Applications". In chapter 1-6 there is the following proof
For any metric space X, there is...
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...
Homework Statement
Show that an open (closed) subset of a metric space E is connected if and only if it is not the disjoint union of two nonempty open (closed) subsets of E.
Homework Equations
The definition of connectedness that we are using is as follows:
A metric space E is...
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...
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...
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...