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 properties of functions and sets that depends on dimension of space.

For $n=1$ (or $m=1$), we have some basic properties of functions and sets that are not valid (or not necessarily valid) for $n\neq 1$ (or $m\neq 1$). For exemple: Calculus. Let $[a,b]$ be a closed ...
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Lipschitz distance

The Lipschitz distance between two metric spaces is defined by $$d_{\mathcal L}(X, Y) = \inf_f \log(\{\max\{\text{dil}(f), \text{dil}(f^{-1})\})$$ where $$\text{dil}(f) = \sup_{x, y \in X} \frac{ ...
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4answers
126 views

How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...
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1answer
26 views

Isometry fixing an open set pointwise is identity

Let $X$ be a metric space and $F$ an isometry of $X$. Suppose $F$ fixes each point of a non-empty open set $U\subset X$. Under what conditions on $X$ does it always follow that $F=\mathrm{id}$? I ...
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13 views

Finsler Metric from page 2 of the book by Chern and Shen.

Physicist here not a mathematician. I am trying to understand the notation for the Finsler metric in Chern and Shen's book. The equation is $$\textbf{g}_y(u,v):=\frac{1}{2} ...
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1answer
67 views

d' is finer than d on compact space $\implies$ $\forall \epsilon \ \ \exists \epsilon' \ \ \forall x \ B_{d'}(x,\epsilon') \subseteq B_d(x,\epsilon)$

This is my conjecture, but I guess I am missing the key idea for the proof (or my conjecture is wrong) Let d and d' be two metrics on a compact space $X$ ($X$ is compact with respect to both ...
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43 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
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1answer
19 views

Lipschitz continuous set-valued map

Let $X$ and $Y$ be metric spaces and $F:X\to2^Y$ be a set-valued map. Suppose that $2^Y$ is endowed with the Hausdorff metric. I wonder about sufficient conditions on $F$ that ensure this map is ...
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60 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
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2answers
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Does a (local) uniqueness theorem exist for geodesics in metric spaces?

In Riemannian geometry, we have the next result. Let $M$ be a smooth manifold with an affine connection. For any point $p\in M$ and for any vector $v\in\mathrm T_pM$, there is a unique geodesic ...
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Show that for any subsets $A,B\subset X$: (i):$d(A\bigcup B)\leq d(A)+d(B)+d(A,B)$ and (ii) $d(\bar A)=d(A)$

Let $d$ be a metric on $X$. Show that for any subsets $A,B\subset X$: (i) $d(A\cup B)\leq d(A)+d(B)+d(A,B)$ (ii) $d(\bar A)=d(A)$ I found this hard to prove because the diameter ...
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In semi linear uniform space

In semi linear uniform space,if f is a function from (X ,ΓX) to (Y,Γy) that is linear and bounded ,is f then continuous? Is the converse true?
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39 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
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1answer
47 views

prove that there is maximum and minimum for C[0,1]

let $C[0,1]$ be the set of continuous functions $f:[0,1]\rightarrow\mathbb R$ and let $A\subset C[0,1]$ be the sub set of twice differentiable functions in $C[0,1]$ which satisfies: $\forall f\in ...
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2answers
57 views

We know that a Compact set is closed. However a finite discreet set is compact but not closed (contradicting the theorem?)

We know that a Compact set is closed. We also know that a finite discreet set is compact (as every cover has a finite sub cover). However a finite discreet set is not closed (contradicting the ...
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20 views

metric-spaces closure

The line in my writing symbolizes the closure, where we add all the boundary points. Is it true that: $\overline{A \cap B}=\overline{A}\cap {B}$? If B is closed, but we do not know if or if not A ...
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49 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
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30 views

Separable metrics

Is the following true?: Let $d_1$ and $d_2$ two separable metrics in space $X$. Then $d=\max(d_1,d_2)$ is a separable metric on $X$. Thanks!
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1answer
34 views

Definition of CAT(0) metric space

I have a question regarding the definition of CAT(0) spaces. I am using the following definition: $X$ complete metric space is CAT(0) if $\forall z,y \in X$, $\exists m \in X$ such that $\forall x ...
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1answer
37 views

Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1 $)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1 $) if they induce equivalent metrics. At first I thought this ...
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Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
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1answer
100 views

Length minimizing curves are geodesic segments

I have a metric space $(X,d)$, a geodesic arc is defined to be a continuous function $\gamma : [a,b] \rightarrow X$, $a < b$, which is (globally) distance preserving and geodesic segments are ...
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$ 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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63 views

Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
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1answer
18 views

Is an open connected subset of Euclidean space a countable sum of open precompact connected subsets?

Let $U$ be an open subset in $\mathbb R^n$. Then there exists a sequence $(U_n)_{n=1}^\infty$ of open precompact subsets of $\mathbb R^n$ such that $U_n \subset cl U_{n+1} \subset U$ and ...
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1answer
23 views

Condition for Lipschitz functions and set inclusion

Let $(X,d)$ be a pseudometric space, and for each $A\subseteq X$ and $\varepsilon \geq 0$ define $$ A^\varepsilon := \{x\in X:\exists a\in A \text{ s.t. }d(x,a)\leq \varepsilon\}. $$ Is this set ...
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How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
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If $f:(A, d_A)\longrightarrow (Y, d^{'})$ and $g:(B, d_B)\longrightarrow (Y, d^{'})$ are uniformly continuous then $h$ is uniformly continuous?

Let $(X, d)$ and $(Y, d^{'})$ be metric spaces, $A, B\subset X$ subspaces such that $X=A\cup B$. Suppose $$f:(A, d_A)\longrightarrow (Y, d^{'})\quad \textrm{and}\quad g:(B, d_B)\longrightarrow (Y, ...
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The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid ...
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How are delta distributions defined in metric spaces?

How are delta distributions defined in metric spaces with continuous metric?
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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 ...
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1answer
64 views

What is the metric spaces needed to motivate concepts of general topology?

I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology? I know it's possible to learn topology without any ...
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1answer
52 views

For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$

Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that ...
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48 views

Qualifier problem: Completeness of Metric Spaces

I am working on old qualifier problems as a review, and I came across this one: Suppose there exists a continuous surjection $f:X_1 \mapsto X_2$, where $(X_1,d_1),(X_2,d_2)$ are metric spaces, such ...
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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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Prove that $[0,1]$ is not isometric to $[0,2]$.

Prove that $[0,1]$ is not isometric to $[0,2]$. Suppose there is an isometry $f:[0,1]\to[0,2]$. Since f is continuous and surjective, the only values for $f(0)$ and $f(1)$ are $f(0)=0$ and ...
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Has anyone seen this space before? Does it have a name?

See the space below (the set taken as a subspace of the plane). It sort of looks like a comb, but with the wrap-around portion added, and the lower left corner removed. What would be a good name ...
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1answer
35 views

Characterization of closed sets in metric spaces

Let $X$ and be $Y$ be two metric spaces and $f:X\to Y$ a continuous function. We know that if $A$ is a closed set in $Y$ then $f^{-1}(A)=\{x\in X, \ \ f(x)\in A \}$ is a closed set in $X$. Now if we ...
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1answer
76 views

If image of closure belongs to closure of image, how to show preimage of interior belongs to interior of the preimage?

Here is exactly what I mean: Define a function $f:X\rightarrow Y$ from a metric space $X$ to another metric space $Y$. If any subset $A$ of $X$ satisfies $f(\bar A)\subset \overline {f(A)}$, then for ...
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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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2answers
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Why does the additive subgroup of $\mathbb{R}$ generated by $1$ and $\sqrt{2}$ contain arbitrary small elements? [duplicate]

Let $G\subset \mathbb{R}$ be the additive subgroup of $(\mathbb{R},+)$ defined by $G=\mathbb{Z}+\sqrt{2}\mathbb{Z}$. I want to prove that for every $\epsilon>0$ there exists an element ...
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1answer
27 views

A proof (result) verification of a statement in metric-spaces

Let $A$ be a set , in a metric-space $(X,d)$ , having no limit points ; then I'm trying to prove that every convergent sequence $(x_n)$ in $A$ is ultimately constant . Please see the proof and tell ...
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1answer
50 views

Completeness of the set of convergent sequences

It's a problem from the book "Topology of Metric Spaces", written by Kumaresan: "Show that the set $\textbf{c}$ of convergent sequences in the Normed Linear Space of all bounded real sequences under ...
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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 ...
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52 views

How to solve for the extrinsic variables of a one variable scaled conformal metric to an equivalent metric?

Given the following metric equivalence \begin{align} e^{2w(x_2)} \left( dx_1^2+dx_2^2 \right) = dy_1^2+dy_2^2+dy_3^2 \end{align} is their a known solution for the extrinsic variables $y_1(x_1,x_2)$, ...
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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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Order of refinement of an open covering of $X$, a metric space

If every finite open covering of a metric space $X$ has a refinement of order $\leqslant n$, is it true that every open covering does too? We say that a covering has order $n$ if $n$ is the largest ...
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Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
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The number of non-degenerate proper subcontinua in a non-degenerate continuum

A continuum is any compact connected metric space. A continuum is non-degenerate if it is not a single point. My question is thus this: How many non-degenerate proper subcontinua must a ...
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1answer
57 views

Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...