Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

learn more… | top users | synonyms (1)

0
votes
2answers
307 views

Distance from a compact subset need not be attained in a metric space?

Suppose we have a metric space $(X,d)$. Let $S$ be a compact subset of $X$. Provide me with an example of $X$, and $S$ (closed and bounded in $X$) such that $$\min \{d(p_0,p): p \in S\}$$ does not ...
0
votes
0answers
17 views

How does one get $p=2$ from a condition that there be non-trivial linear transformations of every dimension that to any power are $p$-norm-preserving?

Verifying that (p=2) satisfies $$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall ...
1
vote
1answer
29 views

Any subset of a metric space is an infinite union of some individual elements of the space?

Let $E$ be a metric space such that the set $\{x\}$ is open $ \forall x \in E$. Does the following proposition make sense? All subsets of $E$ are open. Proof: $\forall S \subset E$, there are ...
2
votes
2answers
30 views

Does every metric on a non empty set can be extended on a super set to a metric?

Let $\phi \ne X \subseteq Y$ , let $d$ be a metric on $X$ , then does there exist a metric $d'$ on $Y$ such that $d(x,y)=d'(x,y) , \forall x, y \in X$ ? What if we also assume that the metric $d$ on ...
2
votes
3answers
40 views

What is the logic underlying this proof?

Proposition: A metric space $X$ is connected if, and only if, every continuous function $f:X\to (\{0,1\},d_D)$ is a constant function, where $d_D$ is the discrete metric on the set $\{0,1\}$. ...
2
votes
1answer
28 views

Prove that two metrics are equivalent

I got stuck on this problem. Hope someone can give some hint to move on. Thanks. Suppose $d_1(x,y) = |x-y|$, $d_2(x,y)=|\phi(x) - \phi(y)|$ where $\phi(x) = {x \over {1 + |x|}}$. Prove that $d_1$ ...
1
vote
1answer
15 views

If an open neighborhood of $x$ has infinite points of $E$, then $x$ is a limit point of $E$

Let $(X, d)$ be a metric space, $E \subseteq X$ and $x \in X \setminus E$. Prove that the following are equivalent: $x \in \overline E$ $x \in \operatorname{Der}(E) = \{x \text{ is an ...
-1
votes
1answer
41 views

$\mathbb{R}^2$ to $\mathbb{R}^1$ Injective Mapping While Preserving the Triangle Inequality

Is there a way to map from $\mathbb{R}^2$ to $\mathbb{R}^1$, such that every point in $\mathbb{R}^2$ has a unique point in $\mathbb{R}^1$ and you preserve the distance (isometry) relations of ...
-2
votes
1answer
68 views

Topological spaces without homeomorphisms?

Can we find a topological space which is not homeomorphic to any other? Of course, not considering the space itself neither the empty set. And if's so, is it possible to classify them? Just like the ...
-1
votes
1answer
20 views

Proper map and sequences in metric spaces

Let $f:X\to Y$ be a continuous map between metric spaces satisfying the Heine-Borel theorem. Show that $f$ is proper if the following condition holds: For every sequence $x_n\in X$ such that ...
1
vote
1answer
32 views

Proving that if $d(x, a) < \varepsilon$ for every $a \in A$, then $d(x, b) \geq \varepsilon$ for every $b \in X \setminus A$

I want to prove the following result: Let $(X, d)$ be a metric space. Then $$\mathring E = \{x \in X \mid d(x, X \setminus E) > 0\}$$ where $d(x, A) = \inf\limits_{y \in A} d(x, y)$. This ...
0
votes
2answers
28 views

Error in proof that the closure of open ball equal the closed ball in all metric spaces

Let $(X, d)$ be a metric space. Denote the open and closed ball as $$B(x_0, r) = \{x \in X \mid d(x, x_0) \lt r\},$$ $$D(x_0, r) = \{x \in X \mid d(x, x_0) \leq r\}.$$ Then $\overline{B(x_0, ...
2
votes
1answer
24 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
0
votes
2answers
35 views

Constructing a metric $\rho$ such that $(\mathbb{R}\setminus \{0\},\rho)$ is a complete metric space

Let $S = \mathbb{R}\setminus \{0\}.$ Construct a metric $\rho$ on $S$ such that (1) $(S,\rho)$ is a complete metric space and (2) for any sequence $\{s_n\}$ in $S$ and $s \in S,$ the ...
1
vote
0answers
15 views

Root distance function in Metric space [duplicate]

Let $\mathbf X = \Bbb R$ with distance function defined by $d(x,y) = {|x-y|}^\alpha$ , where $\alpha \in \Bbb R$ $(0<\alpha\le1)$. Prove that $(\Bbb R , d)$ is a metric space. The first three ...
2
votes
1answer
37 views

Is the space of subsets of $\mathbb R^n$ with the Hausdorff metric separable?

Let M be the Metric Space whose "points" are the Closed and Bounded subsets of a finite dimensional Euclidean Space and whose "distance function" is the Metric defined by Hausdorff for such point ...
0
votes
1answer
22 views

Show that $d_2$ is not a metric.

Show that the function $d_2$ given by $d_2(f_1, f_2)^2 = \int_a^b{(f_1 - f_2)^2}$ is not a metric space on the space of Riemann integrable functions on $[a,b]$. $d_2(f_1, f_2) = 0$ iff $f_1 = ...
2
votes
1answer
36 views

Should a metric always map into $\mathbf{R}$?

Typically you see the definition of a metric as a function which maps $X\times X\to\mathbf{R},$ but does this always have to be the case? Motivating example: When you complete $\mathbf{Q}$ with the ...
3
votes
2answers
63 views

simple proof for principle of pigeons

I must prove the principle of pigeons but the proofs I find in the internet are too complex. Here's what I can use: Definition $$I_n = \{p\in \mathbb{N}; p\le n\}$$ The principle of the pigeons ...
2
votes
2answers
25 views

Example of a bounded space which is not totally bounded

I was trying to find an example of a bounded metric space which is not totally bounded. The only example I could come up whith was the natural numbers with the discrete metric. However, like any ...
0
votes
1answer
15 views

Dense subset in which Cauchy sequences are convergent

Let $S$ be a dense set of a metric space $X$, such that all Cauchy sequences in $S$ are convergent (not necessarily in $S$). Then $X$ is complete space. How can I show that $X$ is complete space ...
0
votes
0answers
21 views

Is this orthogonal distance a common pseudometric?

Define $d: V \times W \to \mathbb{R}$ such that $$d(v,w) = \sup_{z \perp w} \frac{\langle z, v \rangle}{\|v\|\|z\|}.$$ Is this a pseudometric that anyone has utilized in the literature? Does it have a ...
0
votes
0answers
21 views

distribute K points in N dimensional space

I'll try to do my best to simplify the problem, I'm not a Mathematician, I'm a Computer Engineering Student. I'm doing the K-means algorythm, for those who doesn't know what is, is an algorythm to ...
0
votes
1answer
24 views

Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
0
votes
1answer
20 views

Puzzled with this number theory/analysis problem

So, I am having this problem, let $N(x,y)$ be the greatest integer which $b^{N(x,y)}|x-y$ where $x,y$ are integers in $\mathbb{Z}$. Assume that $b \geq 2$. Show $d(x,y)=b^{-N(x,y)}$ is a metric. ...
8
votes
5answers
1k views

Is a ball always connected in a connected metric space?

If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?
1
vote
2answers
35 views

Cauchy sequence of natural numbers

Consider the set consisting of all cauchy sequences $a_n$ with $a_n \epsilon \mathbb{N}$ for all $n$. Is the set countable? My idea: It is straight forward to prove that any such cauchy sequence ...
0
votes
2answers
72 views

Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
0
votes
1answer
25 views

Function of metric with a fixed point

I'm trying to prove that given a metric space $(X, d)$, for a fixed $x\in X$, define the function $g(y)=d(x,y)$, then $g(y)$ is continuous, using triangle inequality. My first question is that can I ...
0
votes
1answer
22 views

Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...
0
votes
1answer
55 views

If X is compact and $C(X)$ is the space of all continuous real valued functions. Prove $C(X)$ is a complete metric space.

Let $X$ be a compact metric space and define $C(X)$ to be the space of all continuous real valued functions on $X$ with a metric defined by $$d(f,g)=\sup_{x \in X} |f(x) -g(x)|.$$ Show that $C(X)$ is ...
0
votes
0answers
5 views

Metric space of non empty closed bounded parts of $R$ with the Hausdorff metric

Consider the metric space of non empty closed bounded parts of $R$ with the Hausdorff metric. For n $\in N_{0}$ and $F_{n} = \{0,1/n,2/n,3/n, ..., 1\}$ i am wondering if $(F_{n})_{n}$ is convergent? ...
0
votes
0answers
18 views

subset of C([0,1]) limited in d1 metric but not in d_inf metric

I am wondering if it is possible to find a subset $C(X) = C([a,b])$ of $C([0,1])$ which is limited for the d1 metric $(d_1(f,g) = \int_{a}^{b} |f(x)-g(x)|dx)$ but not for the $d_{inf}$ metric ...
1
vote
2answers
50 views

Complete metric space

Let $C^0([a,b])$ denote the space of continuous function $f:[a,b]→\Bbb R$. Define $d(f,g)=sup_{[a,b]}|f-g|$. I've proved that d is metric in $C^0([a,b])$. How to prove that this metric space is ...
0
votes
1answer
43 views

A point in a closed set in Euclidean Space [duplicate]

''There exists a point in a closed set which is at minimum distance from a point not in the set.'' I have no idea why this is true. Any help will be appreciated.
0
votes
1answer
15 views

If $d_1$, $d_2$ are metrics on $X$ find a relationship between $\tau_1$ and $\tau_2$.

Suppose $d_1$, $d_2$ are metrics on $X$ and whenever $x_n \rightarrow x$ using $d_1$ we have that $x_n \rightarrow x$ using $d_2$. Let $\tau_1$ be the collection of open sets of $(X,d_1)$ and ...
0
votes
1answer
17 views

Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded?

Suppose $d_1,d_2$ are topologically equivalent metrics on a set $X$. Suppose also that $d_1$ is bounded, that is there exists $K>0$ such that $d_1(x,y) \leq K$ for all $x,y\in X$. Does this mean ...
0
votes
2answers
33 views

Prove this is a metric, what else should I consider?

Let $C_b(\mathbb{R})$ be the space of the bounded continuous functions with values in $\mathbb{C}$ defined in $\mathbb{R}$ ($f:\mathbb{R}\rightarrow\mathbb{C}$) prove that: with $x\in \mathbb{R}, ...
0
votes
2answers
50 views

Metric space and continuity

We define a map $f:(S,d)→(S',d')$ between 2 metric spaces to be continuous at x belongs to S if for every sequence ${x_n}$ in $S$ that converges to x, the sequence {f(x_n)} in $S'$ is convergent to ...
0
votes
2answers
60 views

Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in ...
1
vote
1answer
31 views

Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C_b[0,1]$

Following Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ I would like to prove that the same is true for bounded functions on $[0,1]$ ...
1
vote
1answer
42 views

Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
1
vote
0answers
18 views

How is $d(af(x), af(x_o))$ and $d(f(x), f(x_o))$ related?

I wish to prove that given $f \in C_0([0,1])$ of continuous function, then $af \in C_0([0,1])$ where $a \in \mathbb{R}$ I am having trouble relating $d(af(x), af(x_o))$ with $d(f(x), f(x_o))$ So to ...
-1
votes
1answer
36 views

Is it possible that two metric spaces are metrically isomorphic but not homeomorphic.

I am trying to find an example of metric spaces $(X,d_x)$ and $(Y,d_y)$ such that they are metrically isomorphic, but not homeomorphic. I have been attempting to find one, however I have not been ...
3
votes
4answers
32 views

If $d_1,d_2$ are not equivalent metrics, is it true $(X,d_1)$ is not homeomorphic to $(X,d_2)$?

Consider the statement: If $(X,d_1)$ and $(X,d_2)$ are metric spaces and $d_1,d_2$ are not equivalent metrics, then $(X,d_1)$ is not homeomorphic to $(X,d_2)$. I think this is true, however I can't ...
4
votes
1answer
23 views

There are finitely many $n$-dimensional wallpaper groups

There are $7$ Frieze groups, $17$ wallpaper groups and $230$ space groups, which are the 1,2,3-dimensional cases of isometries on $\Bbb R^n$ (I think? Frieze groups seem to require $[0,1]\times\Bbb R$ ...
0
votes
1answer
30 views

Normalized measure over compact metric spaces

Consider the following definitions. Let $M = (V,T,d)$ be a compact metric space with finite diameter $$D = D(M) = \max d(x,y), ( x, y \in M)$$ and a finite normalized measure $\mu$$M$(.), ...
3
votes
3answers
145 views

Distance from a point to empty set.

Let $(X,d)$ be a metric space and let $A \subseteq X$. We define the distance from a point $x \in X$ to $A$ by $d(x,A)= \inf \{ d(x,a) : a \in A \} $. What will be the value of $d(x, \emptyset )$? I ...
0
votes
1answer
55 views

How can we write (2,5) in the countable family of disjoint open intervals?

I have just read a theorem which states that "Every open subset of R is the union of countable family of disjoint open intervals". Now,I want know how can we write (2,5) in the countable family of ...
4
votes
1answer
28 views

open\closed and disjoint sets under R2

I am stuck with the following question: Consider the sets in $\mathbb{R}^2$ defined by $A = \{(x,1/x)| x > 0 \}$, $B = \{(x, −1/x)| x < 0\}$. Prove that the sets are closed and disjoint, and ...