Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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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, $f(S)\subset Y$ ...
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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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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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$f\colon I\rightarrow G$ 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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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) $S$ ...
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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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Contracting subsets

Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
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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 ...
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102 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, ...
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178 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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271 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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96 views

Is Homeo$(X)$ metrizable?

If $(X,d)$ is a metric space then is Homeo$(X)$ (the group of homeomorphisms of $X$ with itself) endowed with the compact open topology metrizable? At first I thought I could define a metric on ...
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272 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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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 ...
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Are these sets in $\mathbb{R}$ open and/or closed: $\{\frac{1}{n} : n \in \mathbb{N}\}$, $\{0\}\cup \{\frac{1}{n} : n \in \mathbb{N}\}$ and $[0,1)$.

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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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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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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An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min ...
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Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
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+100

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, ...
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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 ...
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+100

Gromov-Hausdorff distance between a line segment and a cylinder

I want to prove the following statement, where $d_{GH}$ denotes the Gromov-Hausdorff distance: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = ...
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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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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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Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
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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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22 views

A specific embedding of semisphere on $R^2$.

I was playing with piece of paper which has the form of semisphere, to be more precise we may assume that it satisfies $x^2+y^2+z^2=1$ for nonnegative $z$. I tried to make it flat without stretching ...
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a weak notion of flow in a metric space

I am seeing the definition of flow in a metric space : $f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$ Note that the condition is ...
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Is $D$ a metric on $X/G$ and does it induce the quotient topology?

Let $(X,d)$ be a compact metric space and $G$ be a finite group of homeomorphisms of $X$. Let $p:X\rightarrow X/G$ be the orbit map. Then we can define a (psuedo) metric on $X/G$ as follows - ...
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Spaces of continuous functions

I have currently been studying space of continuous functions from J.B. Conway's textbook of complex variable. In Chapter VII of the textbook, it uses the notation $C(G,S)$ to denote the space of all ...
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A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
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$E$ 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. ...
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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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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 ...
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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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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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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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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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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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basis of neighborhoods of zero in Schwartz Space

I have the following question: In $S(\mathbb{R}^n)$, and for $\alpha, \beta\in \mathbb{N}^n $, we defined \begin{align} ρ_{\alpha,\beta}(\phi) = ...
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Clarification about a metric

This is rather an easy question but I am a bit confused about the following metric $\rho(\mathcal{G},\mathcal{H}) := \sup_{A\in \mathcal{G}} \inf_{B\in \mathcal{H}} \mu(A \triangle B) + \sup_{B\in ...
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Verify proof to show this is a $\sigma$ - locally finite basis.

Can someone tell me if what I have done is correct? Proposition : Let $X$ be a compact metric space and $G$ a finite group acting on it. Let $p : X \rightarrow X/G$ be the orbit map. Let $B = \{ ...
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Does every manifold M always admit a Riemannian metric?

In the book "Geometry and Topology for Physicists" by Nash and Sen, in Section 7.6, after showing that the structure group $GL(n,\mathbb{R})$ of a frame bundle $F(M)$ (for a general manifold $M$ of ...
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$f$ is continuous, $f : X \to X$, $X$ compact, and $f$ has an $\epsilon$-fixed point for each $\epsilon > 0$. Show $f$ has a fixed point.

Problem: Let $f : X \to X$ be a map from a metric space to itself. A point $z \in X$ is a fixed point of $f$ if $f(z) = z$. Let $\epsilon > 0$. A point $w \in X$ is an $\epsilon$-fixed point of $f$ ...
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Question on complete metric spaces and whether the following is a complete metric space:

Let $ S \subset C^2([0,1])$(set of all two-times differentiable functions on $[0,1]$), which satisfy $$f(0)+f(\frac{1}{2})+f(1)=0.$$ Question :Is $ (S,d)$ is a complete metric space, where $d$ is ...
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Are local quasi-geodesics already quasi-geodesics in hyperbolic spaces?

Recall the following definitions 1) A $(\lambda, \varepsilon)$-quasi-isometric embedding $f$ between metric spaces $X$ and $Y$ is a map $X \to Y$ such that $\frac{1}{\lambda} d_X(x,y) - \varepsilon ...
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Is the inverse of a bijective connectedness preserving map , on a complete real inner product space , also connectedness preserving?

Let $X$ be a complete real inner-product space and $f:X \to X$ be a bijection which maps connected sets to connected sets ; then is it necessarily true that $f^{-1}$ also maps connected sets to ...
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Proving that $S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$ is open in $C^2[0,1]$ with a specific metric

I am trying to prove that $$S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$$ is open in $C^2[0,1]$ with the metric $d$ given by $$ d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ ...
3
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30 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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64 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 ...