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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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 ...
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$f\colon I\rightarrow G$ 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 ...
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90 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
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241 views

compact-open metrizability

Given topological spaces $X$ and $Y$ the set $C(X,Y)$ of all continuous functions $f:X\to Y$ becomes a topological space with the compact-open topology (that is the topology generated by the sets ...
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98 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 ...
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144 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
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157 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 ...
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200 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. $\to$ ...
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187 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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229 views

Are these sets in $\mathbb{R}$ open and/or closed?

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 > ...
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289 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 ...
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43 views

Metric spaces not isometric to any of their proper subsets

Let's say a metric space $X$ has property $P$ if $X$ is not isometric to any of its proper subsets. I'd like to know what this property is called in the literature and whether there's a nice ...
4
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106 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 ...
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60 views

Connected open proper subsets of a connected complete metric space

Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a non-singleton non-empty connected proper open subset?
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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 + ...
4
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89 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
4
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47 views

Metric-like families of relations

Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. ...
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96 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 ...
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106 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
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89 views

lattice of metric structures on a fixed set

Let $X$ be a set. Write $M(X)$ for the set of all functions $d:X\times X\to [0,\infty]$ that endow $X$ with the structure of a generalized metric space (i.e., $d(x,x)=0 $ and the triangle inequality). ...
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51 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
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22 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
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27 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. ...
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37 views

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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23 views

Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
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57 views

Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
3
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67 views

Conditions to make a function a metric on $\mathbb{R}$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$. What conditions ensure that $d(x,y)=|f(x)-f(y)|$ defines a metric on $\mathbb{R}$ Let $g:[0,\infty) \to \mathbb{R}$. What conditions on $g$ ensure that ...
3
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61 views

Topological interpretation of the following equivalence.

We assume $\{X_n\}_{n\in\mathbb{N}}$ and $X$ are random variables from $\{\Omega,\mathcal{F},\mathbb{P}\}$ to $(S,d_s)$, wehre $S$ a separable metric space. One can establish the following ...
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48 views

Sufficient conditions for existence of injection from a metric space $M$ to $\mathbb{R}$

Let $M$ be any metric space. What conditions are required of $M$ for there to exist an injective, continuous function $$\varphi \colon M \longrightarrow \mathbb{R}$$ I would like to believe that ...
3
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189 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 ...
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Can we have an isometric embedding of this metric space into an Hilbert space?

A metric space (from this Q&A), is defined below. I'd like to know if its possible to have an isometric embedding of this metric space into an hilbert space? As per Schoenberg theorem $-d^2(x,y)$ ...
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125 views

Is there a category such that if $\mathbb{R}$ is viewed as an object, we have that $\mathbb{R}^2$ is equipped with the Euclidean distance function?

Viewing $\mathbb{R}$ as an object of the category of metric spaces and metric maps, we have that $\mathbb{R}^2$ is equipped with the distance function $$d(x,y) = ...
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62 views

Metric of space of plane curve

I am looking for a metric $d$ for smooth 2D curves. Hence $d(x,y)$ is the distance between the curves x and y. For the moment, we may assume that $x$ and $y$ are just directed line segments. Do you ...
3
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139 views

Moscow space-Examples

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 $T_1$ ...
3
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46 views

Generalizations of derivatives using distance measures

Let $d(x,y)$ be a distance metric for two points $x,y\in \mathbb{R}^p$. Further, suppose that there are two real or complex sequences $X_n(x)$ and $X_n(y)$, $n=1,2,\ldots$ that depend on $x$ and $y$ ...
3
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378 views

Translation invariant metric

Under what conditions can a metric vector space be given an equivalent metric that is translation invariant? I was wondering if the probability measures on real line can be embedded in vector space ...
3
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127 views

Sequences of Metric Spaces of Compact Subsets

Consider a complete metric space $(M, d)$ and let $F(M)$ denote the non-empty compact subsets of $M$. Then $F(M)$ is also a complete metric space under the Hausdorff distance $d_H$. Given some ...
3
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Lipschitz continuity for an iterated function system

Let $(M,d_M)$ and $(N,d_N)$ be metric and $$ CB(M)=\{\mbox{all closed bounded subsets of }M\}. $$ Let $f: M\to N$ be a Lipschitz map with Lipschitz constant $L$. Define a map $$ F:(CB(M),\rho)\to ...
3
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191 views

Is the Hausdorff semi-distance Lipschitz?

Let $X$ be Banach (with metric $d$) and let $H(X)$ be the set of closed bounded subsets of $X$. Define for $A,B\in H(X)$ $$\delta(A,B)=\sup_{a\in A}\inf_{b\in B}d(a,b)$$ be the Hausdorff semi-distance ...
3
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353 views

Understanding examples - metric spaces, Minkowski functionals and topologies

I'm teaching myself a course on functional analysis but having trouble understanding the notes I've been using. I was hoping I could write out a section of the content and you might be able to help me ...
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79 views

The canonical (1,1)-form on a compact Riemann surface gives locally a subharmonic function

Let $X$ be a compact connected Riemann surface of genus $g>0$. We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ ...
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Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
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57 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
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33 views

Connected $G_\delta$ non-singleton, proper subsets in a connected complete metric space with more than one point

This is a question related to my last; I have still not solved it. Maybe this one is easier: Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a ...
2
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57 views

Zer0-dimensional, countable, 1st countable T1 space is metrizable?

Show that every countable, first countable, zero-dimensional T1 space $X$ is metrizable. I know that T1 space means that all its singletons are closed. Also, zero-dimensional means that $X$ has a ...
2
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176 views

Closure of an open ball equal to the closed ball

If $X$ is a discrete space (metric). Then the closure of a open ball $B_1(x)=\{x\}$ is $B_1(x)=\{x\}$, and the closed ball is $X$, therefore do not coincide. You know another example such that: ...
2
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92 views

Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. ...
2
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77 views

how is this proof about distance in metric spaces is wrong?

Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be $$ d(A,B) = \inf \{ ||x-y|| : x \in A, \; \; y \in B \} $$ For any $A,B$, do we have that $d(A,B) = d( \overline{A}, ...
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34 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
2
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41 views

When does the quotient metric reduce to the infimum of the distances of only two points?

Given a metric space $X$ and an equivalence relation $\sim$, the quotient (pseudo-)metric on $X/\sim$ is defined as follows: $d'([x],[y]) = \inf \left \{ d(p_1,q_1) + d(p_2,q_2) + ... + ...