Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

329
votes
0answers
10k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
151
votes
0answers
3k views

Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
18
votes
0answers
316 views

$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^...
15
votes
0answers
263 views

What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
14
votes
0answers
4k views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then $\...
13
votes
0answers
395 views

Can you build metric space theory without the real numbers?

Every metric space $M$ has a completion $\widehat M$, that is, it can be embedded as a dense subset in a complete metric space. When I first came across this theorem, I thought "well, that's amazing, ...
11
votes
0answers
203 views

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, $\left(-\frac{1}{2},0\...
11
votes
0answers
139 views

Contracting subsets

Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
10
votes
0answers
231 views

Uniform Spaces: Completeness

Attention This thread has been generalized to uniform spaces as general metric spaces. Context The context was the equivalence: $$K\text{ compact}\iff K\text{ totally bounded, complete}$$ That is a ...
9
votes
0answers
244 views

When is an Open Set Homeomorphic to the Interior of its Closure?

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$. I am looking for known assumptions on $X$ and $U$ such that one of the following ...
9
votes
0answers
799 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 ...
8
votes
0answers
293 views

Non empty set with zero diameter

Let $A \subset X$ where $X$ is a metric space. by definition diam$(A) = \sup\{ d(x,y), x,y \in A\}$. if $A$ is non empty and has zero diameter, can I conclude that $A$ is a singleton? i reason as ...
8
votes
0answers
1k views

What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
8
votes
0answers
322 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ It'...
7
votes
0answers
143 views

In which $L^p$ metric is $\pi = 3.5$?

In which $L^p$ metric is $\pi = 3.5$? I am interested because it's well known that $\pi$ can range from $3.14...$ to $4$ in $L^{\infty}$
7
votes
0answers
196 views

Hyperbolic metric spaces 2

I am trying to prove a lemma in Burago's "A Course in Metric Spaces" (Exercise 8.4.4, p.286). Here is a link to a different person's question about the very next exercise in that book, which also ...
7
votes
0answers
821 views

Topological necessary and sufficient condition for tightness

Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$: For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
6
votes
0answers
113 views

A definition of differentiable functions for arbitrary topological spaces

Background It is well-known that there is no notion of derivative for arbitrary topological spaces. However while investigating the notion of derivative as we find in one variable real analysis I ...
6
votes
0answers
70 views

Checking whether $X=\mathbb R$ with $d(x,y)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ is a metric space

Examine whether $d$ is a metric on $X=\mathbb{R}$ where $d\left(x,y\right)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ for all $x,y\in \mathbb{R}$ I think it is not. Even though it satisfies all other properties ...
6
votes
0answers
995 views

If $E$ is a totally bounded subset of a metric space $X$. Then, any subset of $E$ is totally bounded.

Proof: Let $D \subset E$, where $E \subset X$ and ($X,d$) is a metric space. Suppose that $E$ is totally bounded. That is: for all $\varepsilon > 0$, there exist finitely many points $x_1, \ldots , ...
6
votes
0answers
110 views

Is the Hausdorff distance between $I=[0,1]$ and $B_n = \{0, \frac{1}{n}, \cdots, \frac{n-1}{n}, 1\}$ $0$ or $\frac{1}{2n}$?

In this post, it is claimed that the Hausdorff distance $d_H$ between $I=[0,1]$ and $B_n = \{0, \frac{1}{n}, \cdots, \frac{n-1}{n}, 1\}$ equals zero. Is it not equal to $\frac{1}{2n}$ ? For ...
6
votes
0answers
266 views

Find distance to a set (subspace) without computing closest point

General setup: we have a finite-dimensional normed linear space $(V, \| \cdot \|)$, a subspace $U \subset V$, and a fixed vector $v_0 \in V$. We want to find the distance between $v_0$ and $U$. (No ...
6
votes
0answers
210 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
6
votes
0answers
462 views

What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
6
votes
0answers
541 views

Equivalent definition of Cauchy sequence

A sequence $x_i$ is Cauchy if for all $r>0$, there exists $n$ s.t. $i,j\geq n$ implies $d(x_i,x_j)<r$. My question is, is it equivalent to define Cauchy as follows? $x_i$ is Cauchy if for all $...
6
votes
0answers
972 views

Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
6
votes
0answers
364 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have $$d(x,...
6
votes
0answers
3k views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in V_\...
6
votes
0answers
607 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha z_{\bar{\beta}}}...
6
votes
0answers
188 views

Selecting a subset from a set such that a given quantity is minimized

Let $A$ be is a set of some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^A$ denote the mean Euclidean distance from the point $x$ to its $k$ nearest points in $A$ (others than $x$). Let $C \...
6
votes
0answers
264 views

Examples of Moscow spaces

A space $X$ is called Moscow, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G‎_{δ}$‎‎‎-subsets of $X$ . For example, every first countable $T_1$-space is ...
6
votes
0answers
548 views

Are these sets in $\mathbb{R}$ open and/or closed: $\{\frac{1}{n} : n \in \mathbb{N}\}$, $\{0\}\cup \{\frac{1}{n} : n \in \mathbb{N}\}$ and $[0,1)$.

In $\mathbb{R}$, are these sets open? Are they closed? $A = \{\frac{1}{n} : n \in \mathbb{N}\}$ $B = A \cup \{0\} $ $[0, 1)$ My thoughts: $A$ is not open as if we have an open ball with $r > 0$ ...
6
votes
0answers
211 views

Virtually cyclic groups

Let $G$ be a group with finite generating set $A$, define the distance $$d(g,h)=\mathrm{min}\{n:gh^{-1}=a_1^{\varepsilon_1}\dots a_n^{\varepsilon_n}, a_i\in A,\varepsilon_i=\pm1\}.$$ Define the ball ...
5
votes
0answers
44 views

Density of Solution to ODE in Function Space

Let $d$ be a positive integer. Let $f:\mathbb{R}^{d+1}\rightarrow \mathbb{R}^d$ and $g:\mathbb{R}^d\rightarrow \mathbb{R}^d$, be once continuously differentiable functions and define the solution ...
5
votes
0answers
66 views

Is this set dense in $\mathcal{C}^{\infty} (\mathbb{S}^3 , \mathbb{R}^3 ) $?

Consider $\mathcal{F} = {\{ f:\mathbb{S}^3 \to \mathbb{R}^3; f\mbox{ is smooth} \}}$ and the metric $d: \mathcal{F}\times \mathcal{F} \to \mathbb{R} $ $$d(f,g) = |f-g| = \sup_{p\in \mathbb{S}^3} \|f(...
5
votes
0answers
49 views

What does a segment in the plane as a metric space defined by a $p$-norm look like?

In the metric space $\mathbb R^2$ with the metric $d$ defined by $d(x,y)= (|x_1-y_1|^p+|x_2-y_2|^p)^{1/p}$, where $p\gt1$ is a real number, like what does the set of all $m\in \mathbb R^2$ with $d(a,m)...
5
votes
0answers
94 views

Isometrically embed $K$-gon into $\mathbb{R}^N$

In this post, the regular $K$-gon is the metric space $M = \{1,\dots,K\}$ such that, for $k,l \in M$, $$d(k,l) := \begin{cases} |k-l|, & |k-l| \leq K/2, \\ K-|k-l|, & |k-l| > K/2. \end{...
5
votes
0answers
68 views

Functions $f: \mathbb{R}^n \to \mathbb{R}$ such that $|f(x) -f(y)| \le C \prod_{i=1}^n |x_i - y_i|^{\alpha_i}$

The standard definition of a Holder continuous function between metric spaces $X,Y$ is a function $f: X \to Y$ such that there exist $C>0$ and $0 < \alpha \le 1$ such that $$ d_Y(f(x),f(y)) \le ...
5
votes
0answers
486 views

Can we define the derivative of a function in arbitrary metric space in the following way?

Let us first define some terms. Definition of Pre-pseudometric Let $X\ne\emptyset$ and a function $\varphi:X\times X\to\mathbb{R}$ will be called a pre-pseudometric on $X$ if, $x=y\...
5
votes
0answers
101 views

An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min \...
5
votes
0answers
112 views

Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
5
votes
0answers
105 views

Which complete weighted graphs are obtained from finite metric spaces?

Let $(X, d)$ be a finite metric space with $X = \{x_1, \dots, x_n\}$. We can associate to this metric space a complete weighted graph with vertices labelled by the points of $X$, and edges weighted by ...
5
votes
0answers
179 views

Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
5
votes
0answers
409 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. f\...
5
votes
0answers
713 views

Completeness of a metric space with the Hausdorff metric

Let $(Y,d)$ be a metric space and let $K(Y)$ denote the set of all non-empty compact subsets of $Y$. This collection is a metric space when equipped with the Hausdorff distance $h$. I want to prove ...
5
votes
0answers
5k views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
4
votes
0answers
48 views

Is the metric space of Lipschitz Function with $d_{\infty}$ complete?

This is my real analysis homework. Define $$\mathcal{L} = \{f : [a,b] \to \mathbb{R} : f \text{ is a Lipschitz function }\}$$ and the metric $$d_{\infty}(f,g) = \sup\{|f(x)-g(x)| : x \in [a,b]\}$$ ...
4
votes
0answers
44 views

Uncountable collection of open sets.

Can there be an uncountable collection of open sets in $\Bbb R^n$? My idea: Since every open set contains at least one rational number, I can match each open sets to rational numbers and rational ...
4
votes
0answers
37 views

Valid metric on a hyperbolic space

Note: cross-posted to mathoverflow.net I'm looking at the distance that's defined in this paper on Poincaré Embeddings: $d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \...
4
votes
0answers
116 views

Probability distribution to maximize the expected distance between two points

Let $M$ be some metric space. We will also say that $M$ is a measurable space. How do you find a probability measure $P$ on $M$ that maximizes the expected distance between two points? (That is, $P$ ...