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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241 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 ...
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188 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 ...
8
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200 views
Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?
Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb ...
6
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202 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 ...
6
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0answers
124 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
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42 views
Hyperbolic diameter of Amsler's surface
I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of ...
5
votes
0answers
106 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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134 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 > ...
5
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0answers
155 views
Lipschitz continuity of an integral
Let $(E,d)$ be a metric space, $\mathscr E$ be its Borel $\sigma$-algebra and $\mu$ be a $\sigma$-finite measure on $(E,\mathscr E)$. Let the function $p:E\times E\to\mathbb R_+$ be non-negative and ...
4
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0answers
143 views
Metric on the unit cube
Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y,y\neq -x$ define $d(x,y)$ to be the arc length of the path $$Y\cap \{\lambda ...
4
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188 views
Something connected with Arzelà-Ascoli theorem
Let $X$ be a Polish space. Assume that $(C_m)_{m\in\mathbb{N}}$ is an increasing sequence of compact subsets of $X$ and denote $C=\bigcup_{m}C_m$. Let $\{f_n:n\in\mathbb{N}\}$ be a family of ...
3
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0answers
28 views
Is the mapping that takes a metric to the induced intrinsic metric a closure operator?
To abbreviate the expression, "it holds that," I will write "iht."
First a definition. Given a partially ordered set $(P,\geq)$, a closure operator on $P$ is a mapping $\mathrm{cl} : P \rightarrow P$ ...
3
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0answers
50 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 ...
3
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0answers
46 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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0answers
103 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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73 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). ...
3
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0answers
112 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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74 views
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
votes
0answers
106 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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233 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 ...
3
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0answers
69 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 $(\omega_1,\ldots, \omega_g)$ ...
2
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31 views
A sequence of embedded closed balls that have empty intersection
I'm reading soviet textbook "Elements of theory of functions and functional analysis" by Kolmogorov and Fomin. There is an exercise is in it: show example of complete metric space and a sequence of ...
2
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0answers
45 views
convergence in metric space
Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$.
Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
2
votes
0answers
39 views
Regarding nowhere dense subsets and their measure.
A while ago it was made clear that a nowhere dense subset $P \subset [0;1]$ whose Lebesgue measure $\mu(P) = \mu([0;1]) = 1$ doesn't exist.
But is it possible in principle to define a nowhere dense ...
2
votes
0answers
126 views
Space of functions that are everywhere differentiable
Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
2
votes
0answers
55 views
Doubt in Spivak's examples of Manifolds
I've started to study Differential Geometry in Spivak's first volume of his Differential Geometry books. I like very much his approach since general topology isn't assumed, and since he gives many ...
2
votes
0answers
75 views
For a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$, if $F$ is compact on compact subsets of $X$, then $F$ is compact on $X$
The problem as stated in the title isn't quite correct. Let $X$ be a topological space. What I have is a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$ which on compact subsets ...
2
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0answers
142 views
For any point $ a $ of a compact subset $ S $ of a metric space, prove that there exists a nearest point $ c $ to $ a $.
Let $S$ be a compact subset of $X$. Define a metric space $(X, p).$ Prove that for any point $a\in X$, there exists a nearest point $c$ in $S$ to $a$. Moreover, $c$ in $S$ such that $p(c,a)\leq ...
2
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0answers
91 views
convergence of functions on probability measure
I am studying a problem in game theory, but I am lacking on knowledge to deal with a continuum of distribution functions convergence.
$\mathfrak{F}([0,1])$ is the set of distribution functions over ...
2
votes
0answers
46 views
How do I sketch the following metrics:
In $\mathbb{R}^2$ sketch $B((1,2),3)$, the open ball of radius $3$ at the point $(1,2)$, with the following metrics:
a.) the post-office metric given by
$$d(x,y) = \left\{
\begin{array}{l l}
...
2
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37 views
combining pseudo-metrics.
1) Let $D$ be a collection of (uniformly bounded) pseudo-metrics on a set $X$. Then
$$\rho(x,y)=\sup_{d\in D}d(x,y)$$
seems to be a pseudo-metric. Can we embed $(X,\rho)$ in the product space ...
2
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0answers
35 views
How to prove this result about this space of sequences?
Let $s$ denote the metric space of all sequences of real or complex numbers with the following metric:
$$ d( (\xi_j), (\eta_j) ) := \sum_{j=1}^{\infty} \frac{1}{2^j} \frac{|\xi_j - \eta_j|}{ 1 + ...
2
votes
0answers
89 views
How to prove the metric which defined by supremum of all semi-metric?
Define the function $f:X\times X \to R$ by $d(x,y)=\sup\{d_i(x,y):i\in I\}$, when each $d_i$ is a pseudometric; $d_i(x,y)=0$ need not imply $x=y$; for every $i$ in a directed set $(I,\leq)$ and $X$ is ...
2
votes
0answers
54 views
Metric Space Question
Let $(X,d)$ be a metric space and $K \subset X$. $K$ is relatively compact (or precompact) if every sequence $(x_n) \subset K$ has a Cauchy subsequence $(x_{k_n})$. Show that $K$ is relatively ...
2
votes
0answers
73 views
Prove metric space…
Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$
If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by
$$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$$
Prove that $d$ is ...
2
votes
0answers
99 views
Convergence of a function in a metric space to its metric
Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
2
votes
0answers
184 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 ...
2
votes
0answers
69 views
What is the correct distance measure for the (anti) de-Sitter space?
Given these two expressions
1) $\sinh{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1−(t^2−x^2)}}$
2) $\sin{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1+(t^2−x^2)}}$
for distance $d$ from the origin $(0,0)$ to point $(x,t)$, ...
2
votes
0answers
87 views
Inner product and inequalities
Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
2
votes
0answers
70 views
Consequence of metrizability proof - disregard, the question is an error
In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
2
votes
0answers
78 views
What's the relationship between the riemannian metric and Jacobi field?
I encounter to the question in reading the following Excise:
Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...
2
votes
0answers
57 views
Embedding tree metric isometrically into $\ell_\infty$
I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
2
votes
0answers
208 views
How is an integral with respect to a Hausdorff measure defined?
In a reply by Corey:
For integrals of scalar-valued functions on unoriented subsets of $\mathbb{R}^n$, one can use the Lebesgue integral with respect to $k$-dimensional Hausdorff measure ...
2
votes
0answers
147 views
Trouble with some equivalent conditions of compactness
I'm afraid this question may turn out to be a stupid one. Though it is related to a previous question of mine, I'll write it down in full. Let $(X, d)$ be a metric space (MS). I have to prove the ...
2
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0answers
96 views
Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$
I'm looking for examples of spaces $X$ such that:
$X$ is a probability space.
$X$ is a metric space.
If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$.
I ...
1
vote
0answers
15 views
Metric spaces and curvature
Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
1
vote
0answers
19 views
A better way to see this relation concerning Ricci tensor components
If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
1
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0answers
34 views
To show that something is a four-vector
I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
1
vote
0answers
29 views
Is inverse function from metric space to pseudometric space borel?
Let $X$ be a compact metric space and $Y$ a pseudometric space, $f:X\rightarrow Y$ is continuous and bijective.
If there a non-trivial condition that makes $f^{-1}$ Borel?
As Martin commented it is ...
1
vote
0answers
44 views
Determining Complete Metric Spaces
I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$
My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
