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Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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Metric Space defined by an Infinite Sequence of Metric Spaces in this case not a Metric Space

I just started my first book on Topology, Introduction to Topology by Bert Mendelson (Second Edition), and the second chapter is about Metric Spaces. The last problem of Section 2, Problem 2.9, is as ...
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1answer
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Connected metric space of first category

Question: Does there exist a connected metric space of first category (i.e. it can be written as a countable union of nowhere dense subsets)? No such example is given in the book Counterexamples ...
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On the complete metric space

Let $\mathbb N$ be the set of all natural numbers. Set $$d(m, n) = \left\{\begin{array}{ll} 0, &\text{if } m = n, \\ 1+ \dfrac{1}{m+n}, &\text{if }m\neq n. \end{array}\right.$$ Prove that $(\...
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Sequence of points in a nested sequence of sets converges to the point in the limit of the sequence of sets.

Let $B_n$ be a decreasing sequence of compact subsets of a metric space convergent to compact set $B$. That is $B_{n+1}\subseteq B_n$ for all $n$ and $\bigcap\limits_{n\geq 1}B_n = B$. Let $b_n\in B_n$...
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1answer
31 views

How can I prove that my function d(x, y) is a proper metric?

I have points that are characterized by a timestamp and a location, so 3 dimensional points, one temporal x, and two for the location coordinates. My function $d(x, y)$ is defined as follows: $$d(x, ...
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Least possible distance distortion for a map

The background to the question is that I would like to figure out how much a map of Europe must distort distances. Let us try to formulate this mathematically. Say I have a closed subset, $D$, of ...
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1answer
31 views

Gradient and Directional Derivative in Riemannian Manifold

Probably this question is too dumb to be asked, but I am an engineer trying to learn differential geometry, please go easy on me. I am trying to understand that, in Riemannian space, gradient ...
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distance between a two sets in metric space proof [closed]

Let (X,d) be a metric space, A and B be subsets of X so that d(A,B)>0 where d(A,B)=inf{d(a,b):a∈A,b∈B}. Show that, if A and B are compact, then dist(A,B)=d(a,b) where a and b are elements of A and ...
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1answer
41 views

Prove that if a metric space (X,d) is separable, then its completion ($\hat{X}, \hat{d}$) is separable.

Prove that if a metric space $(X,d)$ is separable, then its completion $(\hat{X}, \hat{d})$ is separable. So we want to show there exists a countable dense subset in $\hat{X}$. My attempt: Suppose a ...
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1answer
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A group is a bounded subset of another group

What does it mean when people say that a group is a bounded subset of another group? I need decide whether $SO_2(\mathbb C)$ is a bounded subset of $\mathbb C^{2\times 2}$. What definition should I ...
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Altering the axioms of a metric space

The 3 conditions for a metric space $(X,d)$ are that for all $x,y,z\in X, $ $$d(x,y)=d(y,x)$$ $$d(x,y)\geq0,d(x,y)=0 \iff x=y$$ $$d(x,y)+d(y,z)\geq d(x,z)$$ Are there any interesting results in ...
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Munkres Exercise 39.6. Showing that a collection is countably locally finite but not locally finite.

This is Munkres Exercise 39.6. Consider $\mathbb{R}^\omega$ in the uniform topology. Given $n$, let $\mathscr{B}_n$ be the collection of all subsets of $\mathbb{R}^\omega$ of the form $\Pi A_i$, ...
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1answer
20 views

Intersection in the Hausdorff metric space

Let $(X,d)$ and $(Y,d)$ be complete metric space, and denote by $H(Y,d)$ the Hausdorff metric space on the compact subsets of $(Y,d)$. Let $$ f,g:(X,d)\rightarrow H(y,d), $$ be continuous maps (may ...
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Is $\overline{\langle2\rangle\cdot(4\Bbb Z+1)}=\langle2\rangle\cdot(4\Bbb Z_2+1)$ in $\Bbb Z_2$?

Is $\overline{\langle2\rangle\cdot(4\Bbb Z+1)}=\langle2\rangle\cdot(4\Bbb Z_2+1)$ in $\Bbb Z_2$? $\langle2\rangle$ is the set of powers of $2$ and $\cdot$ is the straightforward dot product. I ...
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1answer
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Existence of Continuous Function respecting points

Suppose that $(X,d)$ and $(Y,d)$ are metric spaces between which there exists a continuous function. Fix $x \in X$ and $y\in Y$. When does there exist a continuous function sending $x$ to $y$?
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Convex Combination of Contraction Maps is a Contraction Map

I'm working on the following problem: Let $K$ be a closed bounded subset of a Hilbert Space $H$ and let $F: K \rightarrow K$ be a nonexpansive mapping $\bigg(d(F(x),F(y)) \leq \alpha d(x,y), \alpha ...
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1answer
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Can one argue homeomorphism via equivalence of metrics? [closed]

Let $S^n$ be the n-sphere with respect to $d_2$ metric (standart metric). Let $C^n$ be the n-sphere with respect to $d_1$ metric. Clearly we have, $S^n$ $\cong$ $C^n$ (homeomorphism). Yet, can one ...
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The metric space (0,1) is closed and bounded, but not compact?

In my lecture notes for one of my modules it states this, but I don't understand how it is closed and bounded, since it's an open interval. The metric space is the open interval (0,1) with the metric ...
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1answer
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Show when $x\neq y$ then in a Metric space $d(x,y):= \epsilon > 0$ [closed]

Let $X$ be a Metric space and $x\neq y$. Show that $d(x,y):= \epsilon > 0$ and I am given as a hint to show that that $B_{ \frac{\epsilon}{2}}(x)\cap B_{ \frac{\epsilon}{2}}(y)=\varnothing$ My ...
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1answer
33 views

Method to construct an open rectangle inside an open ball

I came across this while studying some topology - my school uses the definition of a set $U \subset \mathbb{R}^n$ being open as there being an open rectangle $R$ for all $x \in U$ such that $x \in R$ ...
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How to show that $\text{diam(A)}= \sup \text{rad}(\cdot,A)|_A\leq 2\inf \text{rad}(\cdot,A)|_A$?

$(X,d)$ is a metric space and $A$ is a subset of $X$. The Diameter of a subspace $A$ is defined as $\text{diam}(A):=\sup d|_{A\times A}=\sup_{a,a'\in A}d(a,a')$ And radius of $A$ relative to $x\in X$ ...
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1answer
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Embedding of topological spaces into polytopes: complete regularity and metrizability.

In Dugundji's book Topology an interesting way to study topological spaces shows up frequerently: embedding them into polytopes which are defined by the author as arbitrary cartesian products of unit ...
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contraction map, continuous

Let $(X,d)$ be a bounded and complete metric space. For $f,g:X \rightarrow X$ continuous maps, define $\Delta(f,g)=\sup\limits_{x\in X}d(f(x),g(x))$. Let $\mathcal{C}$ denote the space of ...
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2answers
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Define a sequence $\{y_n\}$ by $y_{2n-1}=x_n$ and $y_{2n}=x$. Which of the following are true?

Let $(X,d)$ be a metric space and let $x_n$ be a sequence in $X$. Let $x\in X$. Define a sequence $\{y_n\}$ by $$y_{2n-1}=x_n,y_{2n}=x,n\in \mathbb N$$ Which of the following statements are correct? ...
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1answer
54 views

Properly discontinuous actions and discrete groups in complete Riemannian manifolds.

I was reading the article "The Geometries of 3-manifolds" by Peter Scott and in the end of page 406 he states the following: If $G$ acts properly discontinuously on a space $X$, then $G$ is a ...
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1answer
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Open and closed round balls in $\mathbb{R}^n$ are convex

I don't know what a round ball is. I hope this is just an unnecessary Detail but if this is important to solve this excercise please let me know. The original text is in German: Offene und ...
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If $f(x)$ is concave can we show that $f(x)= \sup_{y} \inf_{z} \{ f(z): d(z,y) \le d(y,x) \}$ for some metric $d$

Let $V$ be some vectors space and let $f:V \to \mathbb{R}$ be concave. Is it true that for every $x \in V$ \begin{align} f(x)= \sup_{y \in V} \inf_{z \in V} \{ f(z): d(z,y) \le d(y,x) \} \end{...
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1answer
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Space of All Smooth Lipschitz Functions

This is a followup question to this post. Let $F$ be the collection of all Lipschitz functions from $\mathbb{R}^d$ to itself, which admit $k>0$ derivatives everywhere and such that the $k^{th}$ ...
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Can a metric be calculated by another metric?

Let $d: X \times X \rightarrow [0, 1]$ be a metric on $X$ which is computationally heavy to evaluate. I'm interested in the $k$ nearest neighbors of $\alpha \in Y \subsetneq X$ in respect to $d$, ...
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Proving every metric space is normal [closed]

Let $(X,d)$ be a metric space with disjoint, non-empty, closed subsets $A$ and $B$. Define \begin{equation*} f: X \to \mathbb{R}, \ x \mapsto \frac{d(x,A)}{d(x,A) + d(x,B)}. \end{equation*} Show ...
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Railway metric, help showing that the space is bounded but not compact

I am studying metric spaces and I came across this exercice that I was able to solve only partially. Any help and hints will be appreciated. Let $X= \{ x \in \mathbb{R}^2 | \mid \mid x \mid \mid \...
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1answer
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Let $(X,\mathscr T)$ be a metrizable space such that every metric that generates $\mathscr T$ is bounded. Prove that $X$ is compact. [duplicate]

Let $(X,\mathscr T)$ be a metrizable space such that every metric that generates $\mathscr T$ is bounded. Prove that $X$ is compact. My attempt:- We know that $(X,\mathscr T)$ is metrizable. So ...
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Extended metric on the space of probability measures

Let $P(X,\mathcal X)$ denote the collection of probability measures on some measurable space $(X,\mathcal X)$. For $\mu,\nu$ in this space, define $$ d(\mu,\nu)=\log \frac{\operatorname{ess\ sup}_{\...
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Let $f(z)=|z-1|/(|z|^p+1)^{1/p}$, is it true that $f(z.w) \le f(z)+f(w)$?

Let $p \in \Bbb R$ with $p>1$ and $f_p : \Bbb C_{\ne 0} \rightarrow \Bbb R_{\ge 0}$ defined by $$f_p(z)=\frac{|z-1|}{(|z|^p+1)^{1/p}}$$ I'm trying to prove that $$f_p(z.w)\le f_p(z)+f_p(w)$$ I ...
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Is $x \mapsto d(x, A)$ a quotient map?

Consider a metric space $(X, d)$ and a nonempty closed set $A \subset X$. Is the map $d_A : X \to \mathbb{R}, x \mapsto d(x, A)$ a quotient map when restricted to its image? Note $d(x, A) = \inf\{ d(x,...
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Example of a space that is separable but not complete

I know that in general metric space $X$ can be separable without being complete. What's a good example?
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Prove or disprove $bd(E)$ is nowhere dense for $E \subseteq X$ complete metric space

I know this is not true but need to find an example of complete metric space $X$ with subset $E$ such that $\overline{bd(E)}$ has non-empty interior.
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Dense Subsets: If $f|_D = g|_D$ then $f=g$ (proof)

Let $f,g : (X, \rho) \longrightarrow (Y,d)$ be continuous functions and $D$ a dense subset of $X$. Show that, if $f|_D = g|_D$ then $f=g$, where $\cdot |_D$ is the restriction to $D$. Let $w = f -...
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Can such $f$ be continuous at the boundary?

The given problem says , Let $f$ be a complex holomorphic on the open unit disk $D$ such that $|f(z)|\longrightarrow1$ as $|z| \longrightarrow 1$, and $f$ is nonzero inside the open unit disk.Can ...
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Which $[0,\infty]$-categories are compact and connected metric spaces?

The title says it all. Which $[0,\infty]$-categories, regarded as generalized metric spaces, correspond to connected metric space? Which categories correspond to compact spaces? (I have a guess for ...
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The study of Hausdorff distance as a pseudometric.

In hyperspace theory, if $(X,d)$ is a metric space, then the Hausdorff distance between nonempty subsets $A$ and $B$ of $X$ can be defined as $$H(A,B)=\text{max}\{\text{sup}\{d(a,B): a\in A\},\text{...
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1answer
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A continuity problem

$\textbf{Problem:}$ Let $f:(X,d_1)\rightarrow (Y,d_2)$ a continuous function and $B \subseteq Y$. Consider the set : $$ A =\{ x \in X \vert d_2(f(x),Y-B)>0 \} $$ Prove that : $\forall x \in A :...
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Compact + chain-connected metric space => connected

How to show that if $M$ (a metric space) is compact and chain-connected then it is connected ? Definition of $\varepsilon$-chainable : $(X,d)$ is $\varepsilon$-chainable if given any two points $a,b\...
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Heuristic argument for pointwise convergence imply uniform convergence

I didn't want put the statement precisely in the title of the topic to not have a long title, but the items below describe precisely when pointwise convergence imply uniform convergence: Every ...
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2answers
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Find bijection $\ell^{\infty} \to [0,1]$

In a comment on the first answer to this question, @Nate Eldredge stated that "For instance, there is a metric on $\ell^{\infty}$ that makes it isometric to $[0,1]$. How would that look like? ...
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Example on proving continuity in metric spaces

Prove that $f(x)=x+y^2+xy$ is continuous as a function $\mathbb{R}^2\rightarrow \mathbb{R}$ in the metric $d_1(x,y)=|x_1-y_1|+|x_2-y_2|$. (assume that on $\mathbb{R}$ we consider the usual absolute ...
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1answer
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Question on completing proof separability in metric space implies countability [duplicate]

Let $X$ be a metric space. I want to show that: $X$ separable $\Rightarrow X$ second countable. What I have been able to show thus far: Let $Y$ be the countable dense subset in $X$. I have shown ...
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2answers
51 views

Two analysis problems

I am doing these two problems to review my Analysis classes. (a) Show that for each $n \geq 1$, there exists an $x > 0$ such that \begin{equation} \frac{1}{\sqrt{nx+1}} + \frac{1}{\sqrt{nx+2}} + \...
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Intuition on an isometry between the taxicab and max metric in $\mathbb{R}^2$.

I was curious when someone suggested a rotation of $\pi/4$ and a rescaling by $\sqrt{2}$ gives a nice isometry between the taxicab and max metric on $\mathbb{R}^2$. Thus, let $$f: (x,y) \mapsto (\frac{...
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Show that a closed graph and compact codomain implies continuity. [duplicate]

Let $$f:X\rightarrow Y,$$ where X and Y are metric spaces and Y is compact. If f has a closed graph $$G_f=\{(x,f(x)):x\in X\},$$ then f is continuous. My first instinct is to use the fact that if ...