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.

learn more… | top users | synonyms (1)

2
votes
1answer
70 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 ...
2
votes
1answer
50 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 ...
1
vote
1answer
30 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 ...
2
votes
1answer
62 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 ...
0
votes
2answers
31 views

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 ...
1
vote
2answers
50 views

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 ...
0
votes
0answers
14 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
3
votes
1answer
42 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 ...
1
vote
1answer
49 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 ...
1
vote
2answers
60 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 ...
0
votes
1answer
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 ...
0
votes
0answers
51 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 ...
1
vote
1answer
47 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 ...
2
votes
1answer
40 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 ...
0
votes
0answers
16 views

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 $(X_2, d_2)$ ...
3
votes
1answer
126 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 ...
2
votes
0answers
59 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 $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
votes
1answer
64 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 = $\{ z : | z | \leq 1 \}$ ...
0
votes
1answer
21 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 ...
0
votes
1answer
25 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 ...
1
vote
0answers
86 views

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 ...
0
votes
3answers
19 views

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, ...
1
vote
0answers
26 views

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 ...
1
vote
0answers
18 views

How are delta distributions defined in metric spaces?

How are delta distributions defined in metric spaces with continuous metric?
4
votes
0answers
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 ...
1
vote
1answer
71 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 ...
2
votes
1answer
53 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 ...
3
votes
1answer
61 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 ...
4
votes
0answers
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 ...
4
votes
2answers
79 views

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 ...
1
vote
0answers
85 views

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 ...
0
votes
1answer
38 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 ...
0
votes
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 ...
3
votes
0answers
29 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. ...
2
votes
2answers
65 views

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 ...
0
votes
1answer
28 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 ...
3
votes
1answer
55 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 ...
2
votes
0answers
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 ...
0
votes
1answer
54 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 $y_1(x_1,x_2)$, ...
4
votes
0answers
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?
0
votes
0answers
34 views

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 ...
4
votes
1answer
54 views

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 ...
0
votes
0answers
26 views

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 ...
2
votes
1answer
58 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 ...
1
vote
1answer
49 views

Connected $G_\delta$ sets in a connected completely metrizable space with more than one point.

Suppose $(X,\tau)$ is a connected completely metrizable space with more than one point. Let $\mathbb{G}$ be the set of all connected $G_\delta$ subsets of $X$. And let $\mathbb{O}$ be the class of ...
0
votes
2answers
58 views

In a metric space a compact set is closed

I want to show the following: Let $X$ be a metric space. Show that every compact subset $Y$ of $X$ is closed. The idea is to show that $X\setminus Y$ is open. So, for any $x \in X\setminus Y$, I ...
2
votes
1answer
22 views

Local geodesics in uniquely geodesic spaces

Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a ...
4
votes
1answer
61 views

Do connected complete metric spaces always contain a path?

Does every connected complete metric space with more than one point contain a non-trivial path? The pseudo-arc is an example of a connected metrizable space without a path.
3
votes
1answer
35 views

Connected complete metric spaces with more than one point.

Does every connected complete metric space with more than one point have infinitely many closed balls? And is any closed ball in a connected complete metric space connected?
0
votes
2answers
46 views

if $A, B$ are open in $\mathbb R$ then so is $A+B.$

I am trying to find out a counterexample to the problem: if $A, B$ are open in $\mathbb R$ then so is $A+B.$ But I could not find any such counterexample. Please help me.