Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Interesting Metrics

To preface, this question might come off as a bit silly, but it would serve me well to understand the concepts, and to actually get a good answer for this. How can I design an ideal metric for ...
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40 views

Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
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Verifying the triangle inequality for a metric for hyperbolic space

I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...
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81 views

Completeness in a Category of Metric Spaces.

Is there a way to describe completeness within a category of metric spaces? The point is that I'd like to have a description of compactness in metric spaces by something of the form totally bounded + ...
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1answer
57 views

Subset of infinite connected set

How to proove that infinite connected set has got proper infinite connected subset?
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39 views

How to prove that a metric space is compact if it is complete and totally bounded?

How to prove that a metric space is compact if it is complete and totally bounded? Wiki wrote that it is a generalisation of Heine–Borel theorem but I can't prove it.
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3answers
41 views

Which of the following sets are dense in $C[0,1]$

Which of the following sets are dense in $C[0,1]$ with respect to sup-norm topology? $1$. {$f$$\in$ $C[0,1]$ : $f$ is a polynomial } $2$. {$f$$\in$ $C[0,1]$ :$f(0)$=$0$} $3$. {$f$$\in$ $C[0,1]$ ...
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3answers
273 views

Is distance between two sets equal to that between their boundary?

I am not sure if the statement below is true. The statement is: Let $(M,d)$ be a connected metric space and $A, B$ be two nonempty subsets of $M.$ Assume the boundary $\partial A$ and $\partial B$ are ...
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1answer
29 views

A question about closed ball in metric space

Question: Let $(M,d)$ be a metric space and $\Omega$ be a bounded open subset of $M.$ For every positive real number $\epsilon,$ let $$\Omega_{\epsilon}:=\{x\in\Omega \mid ...
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1answer
33 views

Is the diameter of intersection of a set with a sphere of radius $r$ a measurable function of $r$?

I have to face to following problem: let $X$ be a separable metric space and $x_0 \in X$ fixed. Consider an open bounded set $A \subset X$. I want to know if the function $f: [0, \infty) \mapsto [0, ...
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1answer
31 views

Can someone criticise my incorrect proof about a set being open?

In the question I have to decide whether the set $S=\{(x,y)\in\mathbb{R}^2\;|\;x/y\leq 7\}$ is open, closed or neither. I attempted to prove it was closed but it turns out it is neither can someone ...
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1answer
15 views

$l_{p}$ metric on $\mathbb{R}^{n}$ and its open balls

For $x,y \in \mathbb R^n$ let $$ d_p(x,y) = \left(\sum_{i=1}^n \def\abs#1{\left|#1\right|}\abs{x_i - y_i}^p\right)^{1/p}$$ for $1 \le p < \infty$ and $$ d_\infty(x,y) = \max\{\abs{x_i -y_i} ...
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2answers
30 views

triangle inequality to show metric

$d(x,y)= \begin{cases} 0 &\mbox{if } x=y \\ 1+\frac{1}{x+y} & \mbox{if } x\neq y \end{cases} $. Show that $(\mathbb{Z}^+,d)$ is a metric space. I'm stuck in proving triangle inequality.
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Help Me Understand: Proof that Finite Intersection of Open Sets is Open

The proof is here: (link). I don't see how the third line (starting with Thus: $\exists \epsilon_i$...) is justified. That is: just because $x \in U_i$, for all $i$, how do I know that a ...
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2answers
24 views

replace convergence with continuity?(metric spaces)

This question is convcerning metric-spaces. In theory we can replace continuity with convergence. That is, since continuity in a point a is equal to the statement that if $\{x_n\}$ is any sequence ...
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1answer
11 views

A question about equivalent metrics.

I'm trying to prove that if the convergent sequences of $(X,d)$ and $(X,\rho)$ are the same, then the metrics $d$ and $\rho$ are equivalent. Equivalent metrics are those that generate the same open ...
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1answer
14 views

Net-Complete $\iff$ Sequence-Complete

Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets! (The converse is trivial of course!)
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1answer
42 views

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz. Is the above statement true? I thought since $f$ is continuous on a compact metric space, $f$ ...
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3answers
56 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
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1answer
33 views

Can I show these questions (is a set open or closed WRT metric) a faster way?

I have the metric: $$d((x,y),(a,b))=|y-b|\text{ if }a=x\text{ else }|y|+|b|+|x-a|$$ I have been asked the following questions: Is the set $\{0\}\times(0,1)$ open with respect to this metric? Is it ...
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1answer
35 views

Difference between F-space and Frechet space in W. Rudin's “Functional Analysis”

In Walter Rudin's book, "Functional Analysis", we read that by talking about local base, he will be thinking about neighborhoods of $0$. In the vector space context, the term local base will ...
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1answer
35 views

int$(A) \subseteq$int$(A')$ and int$(A) \subseteq A'$

In which metric spaces is it true that int$(A) \subseteq$int$(A')$ ? (I know it is true in $\mathbb R$) Moreover in which metric spaces is it true that int$(A) \subseteq A'$ ? (I know it is true in ...
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1answer
21 views

“Every seq. in $X$ has a Cauchy subseq.” implies “$\forall\epsilon > 0, \exists $ a finite set $T$, s.t. $\forall x\in X, d(x,T)<\epsilon$.”

I have a proof of the following theorem. Let X be a metric space. "Every sequence in X has a Cauchy subsequence" implies that "$\forall\epsilon > 0, \exists $ a finite set T, s.t. $\forall x\in X, ...
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1answer
27 views

Function that's a metric on one space but not another?

Is there a function which makes sense on two sets and is a metric on one but not the other? I can't seem to come up with an example or a proof a metric on one set implies it is on every other one it ...
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31 views

For any countable $ A$ , $B \subseteq A \implies B \cap B\space' \ne B $

In which kind of metric spaces is the following true For any non-empty countable set $A$ of the metric space , $B \subseteq A \implies B \cap B\space' \ne B $
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1answer
124 views

Hyperspace and connectedness

I'm looking for any theorems and proofs for connectedness for hyperspaces exp(X). I would like to take a look for especially this theorem: $$ X \textit{ is connected } \leftrightarrow exp(X) ...
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2answers
71 views

Question on two metric spaces properties

Question: Let $X$ be a set and let $d_1$ and $d_2$ be two metrics on $X$. Assume that there exists a constant $C > 0$ such that $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...
2
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3answers
56 views

Metric Space, Induced topology

I'm trying to show that the metric topology is indeed a topology. To do so, I want to show the following three statements are true: $\emptyset$ and X are in $\tau$ finite intersections of open sets ...
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1answer
68 views

Is sum of two metrics a metric?

The production of two metrics is a metric also. It's googled easy. But what's about a sum? As I can see sum is metric, as the triangle inequality of metric sum is the consequence of the inequality ...
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1answer
28 views

Let X be a complete metric space in which every closed ball is uncountable. Prove that X has cardinal number greater or equal than the continuum

Let X be a complete metric space in which every closed ball is uncountable. Prove that X has cardinal number >= c (continuum) (Can you please prove with properties of Separability of a Metric Space? ...
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1answer
32 views

Cauchy inequality proof

I am studying cauchy inequality proof from notes I have from my class$$(\forall\vec{x},\vec{y}\in\mathbb{R}^n):|\sum_{i=1}^{n}x_iy_i|\le||\vec{x}||\cdot||\vec{y}||$$ We choose $\vec{x},\vec{y}$. And ...
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1answer
42 views

Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
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1answer
14 views

Gaussian kernels for arbitrary metric spaces

Let $(I,d)$ be an arbitrary (pseudo-)metric space. Define the function $$c(i,i') := \exp\big( - d(i,i')^2 / 2 \big)$$ Is $c$ necessarily nonnegative-definite, hence a kernel function?
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30 views

Is this subset of a finite metric space already named?

Given a finite set, $X$, with a metric, $d(x,y)$ defined on it, I am interested in the following subsets: $S_k\subseteq X$ s.t. $\forall x\in X,\exists s\in S_k:d(x,s)\geq k$ Do such constructions ...
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56 views

Examples of metric spaces in which every non-empty open set is uncountable

Is every non-empty open set of a complete metric space uncountable ? If not can anyone please provide some examples of metric spaces (other than $\mathbb R$ with usual metric) in which every non-empty ...
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1answer
31 views

Is the Set of Distances Between a Finite Open Subset and a Closed Subset of a Metric Space Closed?

In order to be as clear as possible, I've taken the liberty of TeXing (Tikzing?) up the sort of image in question. Here, $\gamma$ is some path in the complex plane, the disk ...
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1answer
81 views

Complete metric subspaces of $\mathbb{Q}$

Is there a nice characterization for the complete metric subspaces of $\mathbb{Q}$ (with the usual metric)? It seems like a such a subspace must have empty interior; if it contained an open interval ...
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1answer
44 views

Cardinality of all compact metric spaces

I`m looking for cardinal number of all compact metric spaces. I know that: Cardinal number of compact set is at most $\mathfrak{c}$ (it is a continous image of Cantor set) Compact metric space is ...
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1answer
48 views

Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
0
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1answer
21 views

Continuous function in metric spaces

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. $f(x): (P,\rho) → (\mathbb R,\rho_1): 0$ if $x∈[-1,1]$ or $1$ if $x∈\mathbb R/[-1,1]$. $\rho_1 (x,y) = \sum_{k=1}^\infty |x_k-y_k|$. ...
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1answer
34 views

Non-constant Cauchy sequence

I need to find an example of non-constant Cauchy sequence in $\mathbb E^2$. The metric in question is $\rho_2$, so Cauchy sequence would be sequence for which following is true: $\sqrt {(x_m - y_m)^2 ...
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1answer
51 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
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1answer
22 views

Prove that the distance function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ $0<p<1$ is a metric on R^n

Hi I am trying to prove that for $0<p<1$ the function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ is a metric on $\mathbb{R}^n$. I am struggling with the triangle inequality part; We have to prove ...
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1answer
89 views

A continuous function on the real line such that the preimage of every point is either empty of has exactly 3 points

Let $f : \mathbb{R} \to \mathbb{R}$ be a function with all fibres $(\lbrace{x \in \mathbb{R}| f(x) = c\rbrace} = f^{−1}(c)$, $c \in \mathbb{R})$ either empty or consisting of exactly three points. ...
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1answer
21 views

Proving that $b \in \overline{A}$ if and only if $\rho(b,A) = 0$

I need some help with this problem: Let be $(X,\rho)$ a metric space, $A \subseteq X$ and $b \in X$. The distance from $A$ to $b$ is defined as $\rho(b,A) = \inf\,\{ \rho(b,a) : a \in A \}$. Prove ...
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Sufficient conditions for RTree

What is the sufficient screening criteria of a space for the possibility to use R-Tree spatial index on it? I cannot apply it to a space with just Jaccard distance as the metric. As I suppose the ...
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1answer
46 views

Finitely many connected components, prove interiors are also connected

Show that in a space with finitely many connected components $C_i, i = 1, ..., n$ their interiors $Int(C_i)$ are also connected. Is it true in general that the interior of a connected component is ...
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1answer
27 views

Showing a metric space is complete.

On the space of continuous functions on $[0,1]$, I have a metric $$d(f,g) = \sup | \alpha(x) (f(x) -g(x))|,$$ where $\alpha(x)$ is a continuous function and $\alpha(x) \ne 0$. I'm trying to find ...
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1answer
45 views

Group of distances

How to prove that $$g:\Bbb R^3 \to \Bbb R^3 \in G = \{g \, | \, \text{ for each } g \text{ exists } n\in\mathbb{Z} : r(g(x),g(y)=2^n r(x,y) \}$$ for each $x,y\in \Bbb R^3, r$ is an euclidean ...
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1answer
82 views

Equivalent of $\ln\ln(N(\epsilon))$ where $N(\varepsilon)$ is the minimum of balls for covering $A$.

Let $E=\mathcal{C}^0([0,1],\mathbb{R})$ with the uniform convergence and $$A=\{f\in E\ |\ f(0) = 0\text{ and } \forall x,y\in[0,1]\ |f(x) - f(y)| \leqslant |x-y|\} $$ For $\varepsilon >0$ ...