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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Completion of a metric space in categorical terms

Is it possible to define the completion of a metric space using categorical terms?
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1answer
3k views

How to prove triangle inequality for $p$-norm?

Well, I've been studying metric spaces and to make the cartesian product of metric spaces a metric space I've heard of the $p$-norm defined in $\mathbb{R}^n$. So if $\mathcal{M}=\{M_i : i\in I_n\}$ is ...
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How to show that the distance between these sets is positive?

Let $T_i=\{(1-t)x_i+ty_i;\;0\leq t\leq1\}$, where $x_i,y_i\in\mathbb{R}^n$; $i=1,2$. Could someone help me to prove that if $x_2= (1+\varepsilon)x_1$ and $y_2= (1+\varepsilon)y_1$ for some ...
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1answer
249 views

Clustering of Cofinally Cauchy nets

If $(X,d)$ is a metric space in which every Cofinally Cauchy sequence clusters. Does this imply every Cofinally Cauchy net clusters in the space?
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1answer
119 views

A result about connectedness and closed set.

Show that if $F$ is a closed and connected subset of a metric space $X$ then for every pair of points $a,b\in F$ and each $r>0$ there are points $z_0,z_1,\ldots,z_n$ in $F$ with $z_0=a$, $z_n=b$ ...
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165 views

Distance metric for (infinite) lines in 3D

I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties: If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = ...
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1answer
60 views

One Point Derivations on locally Lipschitz functions

Let $A$ be the algebra of $\mathbb{R}\to\mathbb{R}$ locally Lipschitz functions. What is the vector space of derivations at $0$? The proof that for continuous functions there aren't really any doesn't ...
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1answer
142 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
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1answer
103 views

How to describe the family $\tau$ of all open sets of $(\mathbb R^2,\delta)$

Ex. 1.2.65. Let $d$ be the Euclidean metric on $\Bbb R^2$. Define $$\delta(p,q):=\begin{cases} d(p,0)+d(q,0), & p\ne q \\ 0, & p = q, \end{cases}$$ for $p, q \in \Bbb R^2$. Show that ...
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1answer
89 views

geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
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378 views

Baby Rudin 2.26 Infinite subsets with limit points implies compactness

Having some trouble with this question. Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. Hint: By Exercises 23 and 24, $X$ has a ...
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5answers
137 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
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5answers
166 views

Find a metric space X and a subset K of X which is closed and bounded but not compact.

Find a metric space $X$ and a subset $K$ of $X$ which is closed and bounded but not compact. I can find a metric space $X$ like the below. Let $X$ be an infinite set. For $p,q\in X$, define ...
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0answers
70 views

Relationship metric space and $\sigma$-discrete base

Hy, I am newbie here. Can you help me to prove this proposition? If $X$ metric space, then there is a $\sigma$-discrete base $\mathcal{U}$ for the topology of $X$, i.e., ...
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1answer
58 views

I couldn't find the fault in $B_X(a,\epsilon)\times B_Y(b,\epsilon)=B_{X\times Y}((a,b),\epsilon)$

I know that the product of two balls of equal radius in metric spaces is not necessarily a ball in the product space. But I couldn't identify the fault in the proof where I showed ...
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1answer
66 views

A definition of metric space

Can you please help me solve the question below? I have no idea how to prove this one. Define the set $$X:=\{K\subset\mathbb C:K\text{ is bounded and closed}\}$$ Define a function $d\colon X \times X ...
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208 views

Metric Space (Elementary Analysis)

Let $d: X \times X \to \Bbb R$ is a function satisfying all properties of a metric space but $d(x,y)=0 \implies x = y$. If we define $\sim$ on $X$ by $x\sim y \iff d(x,y) = 0$, prove that $D([x], ...
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1answer
212 views

Lebesgue's criterion for Riemann integrability of bounded real valued functions defined on compact metric spaces

Let $(X, d)$ be a compact metric space, and let $S$ be the algebra of sets generated by the open and closed balls of $X$. Suppose we have a pre-measure defined on $S$ such that the measure of each ...
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1answer
124 views

I have to determine which of the following define a metric on $\Bbb R \,\,$?

I am stuck on the following problem: Determine which of the following define a metric on $\Bbb R$: $d(x,y)=\frac{|x-y|}{1+|x-y|}$ $d(x,y)=|x-2y|+|2y-x|$ $d(x,y)=|x^2-y^2|$ ...
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1answer
338 views

Examples of homeomorphisms between the real numbers and the positive real numbers?

I'm interested in homeomorphisms between the real numbers, $\mathbb{R}$, and the positive real numbers, $(0,\infty)$--where both spaces have the topology induced by the metric $d(x,y)=|x-y|$. Here ...
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2answers
187 views

Is there a structure theorem for nonempty, compact, nowhere dense subsets of the real line?

Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line. Is there a theorem that describes the form of the elements of $X$? Context For open subsets of the line, such a ...
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233 views

why we want to use grassmannian space?

I wonder what's the special about grassmannian space? Why we want to use this space? On wikipedia, it says: "By giving a collection of subspaces of some vector space a topological structure, it is ...
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1answer
131 views

Prove triangle inequality

I want to prove that $d(x,y) = 1- \sum_i {\min(x_i, y_i)}$ where $\sum_i {x_i} = \sum_i {y_i} =1$ and $\forall i: x_i, y_i \geq 0$ satisfies the triangle inequality. The domain of $d$ therefore is ...
4
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1answer
114 views

Topologies coinciding at a point or a set.

Consider a set equiped with two topologies. What does it mean to say that the two topologies coincide at a point in the set? Is it meaningful to talk about this concept in general. Is it meaningful in ...
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2answers
198 views

Why is the discrete metric said to be so important

Can anyone enlighten me as to why the discrete metric is considered to be important in mathematics? The only real use I can see of it is that it shows the existence of a metric on any non-empty set. ...
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1answer
53 views

Extension of a metric defined on a closed subset

If $X$ is any metrizable space, $A$ is a closed subset of $X$. Let $d$ be a compatible metric on $A$ then $d$ can be extended to a compatible metric on $X$.
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2answers
356 views

When is a metric space Euclidean, without referring to $\mathbb R^n$?

Normally, the Euclidean space is introduced as $\mathbb R^n$. However, I've now been thinking about how one might define the $n$-dimensional Euklidean space only from the properties of the metric. ...
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2answers
192 views

Metrizability of a compact Hausdorff space

Show that a compact Hausdorff space is metrizable if the diagonal in $X\times X$ is a zero set of a non negative function ?
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1answer
75 views

Star graph embeddings

This is an homework question which I'm struggling with: Let $S = (V, E, w)$ a star graph, meaning, $S$ is a tree that all it's vertices are leafs except one. I need to : show that every weighted ...
2
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1answer
128 views

Cluster point of a sequence $\{x_n\}$ is the limit of some subsequence - Axiom of Choice? [duplicate]

In a metric space, a cluster point of a sequence $\{x_n\}$ is the limit of some subsequence. The only proof that I know works like this: Construct a sequence $\delta _k \to 0$. For each $\delta _k$ ...
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0answers
78 views

Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
14
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1answer
191 views

a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
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299 views

Dual and completion of metric spaces

Say we have a metric space $(M,d)$, and we want to complete it in the following sense: Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
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1answer
220 views

Pre-compact balls of a separable metric space

Let $(X,d)$ be a complete metric space (I am not assuming that the metric is finite, there could be points in $X$ with infinite distance). Assume that each Ball in $X$ is pre-compact i.e. $\forall x ...
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2answers
293 views

Is the identiy function continous on equivalent metric spaces

Let $f$ be the identity function from $(X,d_1) \to (X,d_2)$. If $d_1$ and $d_2$ are equivalent metrics we can deduce that the identity function is continuous, right? Since for every open set $G$, ...
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2answers
125 views

$d$ is a metric on $X$ if $d(a,b) = 0 ⇔ a = b$ and $d(a, b) ≤ d(z, a) + d(z, b)$

The following is a question of Metric Spaces by O'Searcoid (pg 19) Suppose $X$ is a set and $d:X×X→\mathbb{R}$. Show that $d$ is a metric on $X$ if, and only if, for all $a,b,z ∈ X$, the two ...
2
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2answers
289 views

Equivalence relation on the collection of metric spaces.

Ok, to start I am new to metric spaces. I have studied equivalence relations in Algebra, but unfamiliar with the e.r. in metric spaces. Here is my question: We say that metric spaces (X,$d_X$) and ...
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1answer
33 views

A statements on metric spaces

Need to verify a few statement regarding metric spaces, found them in a few questions Let $(X,d)$ be a metric space 1) $X= \bigcup_{n\in \Bbb N} B(a,n) $ where $B(a,r)$ indicates the open ball with ...
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0answers
701 views

Showing the Unit Circle is Connected

One way to show that the unit circle is connected is to use the map $f: [0, 2\pi] \to \mathbb{R}^2$ where $f(x) = (\cos x, \sin x)$. Since $f$ is a continuous map and $[0, 2\pi]$ is connected, the ...
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1answer
31 views

Existence of a statistical distance with a special property

Let $A, B, C, \ldots$ be a set of points. Is there a non-trivial or artificially cooked up statistical distance which satisfies the following conditions: The nearest point to $A$ is $B$ Leaving ...
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1answer
127 views

Which of these statements regarding metric spaces are true?

The following are a few statements in various metric spaces mcqs that I couldn't figure if they are true or false. Please offer some help to get answer them Let $(X,d)$ be a metric space 1) If $ ...
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3answers
73 views

$\overline{A}\cap B\neq\emptyset$ or $\overline{B}\cap A\neq\emptyset$ implies $\operatorname{dist}(A,B)=0$

$\overline{A}\cap B\neq\emptyset$ or $\overline{B}\cap A\neq\emptyset$ implies $\operatorname{dist}(A,B)=0$. I have tried to prove this but was unable to find a decent method. Any help will be ...
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4answers
728 views

Why does Totally bounded need Complete in order to imply Compact?

Why does "Totally bounded" need "Complete" in order to imply "Compact"? Shouldn't the definition of totally bounded imply the existence of a convergent subsequence of every sequence?
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1answer
112 views

Pair wise disjoint open balls

Let $(X,ρ)$ be a metric space and $A$ is a subset of $X$. How do I prove that there exists a family $(B(t,r_{t}))_{t\in\operatorname{Iso}A}$ of pair wise disjoint open balls?
2
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1answer
144 views

Embeddings of 3 point metric spaces into ultra-metric spaces with distortion 2

I need to show that every 3 point metric space has an embedding into an ultra-metric space with distortion 2. And then to show such an example. How would I go about it? Thank you. Edit: Distortion ...
3
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2answers
120 views

Products and Stone-Čech compactification

Let $X$, $Y$ be complete metrizable spaces, $\beta X$, $\beta Y$ be their Stone-Čech compactifications. It is known that $C_b(X) \simeq C(\beta X)$. Is it possible to say something about the relation ...
3
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3answers
2k views

A metric space is separable iff it is second countable [closed]

How do I prove that a metric space is separable iff it is second countable?
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76 views

Metric on a set

Can someone provide a hint for solving the following. Show that $d:(R^{\infty})^2\to R_+$ is a metric. $$d(x, y)=\sqrt{\sum_{i=0}^{\infty}{(x_i-y_i)^2}}$$ I need a hint for showing that $d$ ...
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2answers
67 views

A problem on distance of sets

Let $(X,d)$ be a metric space and $ A,B,C$ be non empty subsets of $X$. If $\mathrm{dist}(A,B) = \inf \{d(x,y) \mid x\in A , y \in B \}$, show that $\mathrm{dist}(A,C) \le \mathrm{dist}(A,B)+ ...
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0answers
24 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...