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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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, $f(S)\subset Y$ ...
49
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975 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 ...
13
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241 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 ...
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150 views

Global structure of the Gromov-Hausdorff space

EDIT: now crossposted at mathoverflow (http://mathoverflow.net/questions/212364/on-the-global-structure-of-the-gromov-hausdorff-metric-space) This is a purely idle question, which emerged during a ...
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124 views

Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ ...
8
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125 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 ...
8
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367 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 ...
7
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88 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, ...
7
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148 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 ...
7
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268 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. $\to$ ...
6
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228 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 ...
6
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159 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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500 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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166 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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48 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
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143 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 ...
5
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321 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 ...
5
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319 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 > ...
4
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43 views

Does a homogeneous metrizable space admit a compatible homogeneous metric?

Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ...
4
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40 views

$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...
4
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72 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 $d$? ...
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59 views

For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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74 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
4
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42 views

$\epsilon$-isometry of a compact metric space is $\epsilon$-surjective

The question whether an isometric map $f : X \to X$ of a compact metric space is surjective has been asked (and answered positively) frequently. Assume more generally that $\vert d(f(x),f(y)) - ...
4
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65 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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123 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. ...
4
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69 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?
4
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817 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 ...
4
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51 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. ...
4
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221 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 ...
4
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143 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 ...
4
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96 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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18 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
3
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60 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
3
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39 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
3
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34 views

Example of a metric group over $\mathbb{R}_0^+$

Do you know an example of a function $d:\mathbb{R}_0^+\times\mathbb{R}_0^+\to \mathbb{R}_0^+$ for which the following properties hold? Or can you prove this does not exist? There exists an $e\in ...
3
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35 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
3
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57 views

When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
3
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35 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?

My textbook says the following: Let $X=[1,\infty)$ and define $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac1x-\frac1y|$ . Then $d_1$ and $d_2$ are inducing the same topology, $(X,d_1)$ is complete but ...
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40 views

To Show that $(\Bbb R,d)$ is bounded and complete but not totally bounded.

On the set $\Bbb R$ of reals consider the metric $d$, given by $d(x,y) = min \{ 1, |x-y| \}$. To Show that $(\Bbb R,d)$ is bounded and complete but not totally bounded. Bounded can be easily verified ...
3
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To prove that $f(A)$ is compact in $(Y,e)$.

Let $(X,d)$ and $(Y,e)$ be metric spaces, $A \subset X$ is compact and $\eta$ a fixed number and $f : A \to Y$ a function such that $$e(f(x),f(y)) \leq \eta d(x,y) \ , \ \forall x,y \in A$$ To prove ...
3
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52 views

Assume that the metric space $(X, d)$ is not compact then there exist a $f: X \to \Bbb R$ which is continuous but not bounded.

Assume that the metric space $(X, d)$ is not compact then there exist a $f: X \to \Bbb R$ which is continuous but not bounded. I am finding difficulty in constructing such a function!!
3
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53 views

A Theorem On Compact Connected Metric Spaces by Stadje

I recently came across a surprising theorem, due to Wolfgang Stadje, a special case of which states that: Let $(X,d)$ be a compact connected metric space. Then there exists a unique real number ...
3
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35 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 ...
3
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64 views

Gelfand width of $l_1$ unit ball in $\infty$ metric

The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not ...
3
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27 views

How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...
3
votes
0answers
143 views

Problem 1, Chapter 3 - A Comprehensive Introduction to Differential Geometry - Spivak

There seems to be an error in item (d) of Problem 1, Chapter 3 of Spivak's A Comprehensive Introduction to Differential Geometry, 3rd Edition, which I transcribed below. There is something wrong with ...
3
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44 views

Showing equality of sets in $C[a,b]$

The exercise states: Let $a,b\in\mathbb{R}$, $a<b$ and let $(C[a,b],\Vert\cdot\Vert)$ denote the vector space of continuous real functions on $[a,b]$ endowed with the uniform norm. Let ...
3
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113 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 ...
3
votes
0answers
32 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 ...