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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Can we classify all topological spaces where separability of every subspace implies the the topological space is metrizable?

$(X, \tau) $be a topological space and $(Y, \tau_Y) $ be it's topological subspace. I know if $(X,\tau)$ is separable and metrizable then $(Y,\tau_Y)$ is separable. Suppose $(Y, \tau_Y) $ is separable ...
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Show that $(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$ is continuous

$d_\infty = \sup |f(x) - g(x)|) \ $ and $ \ d_1 = \int_{i=1}^n |f(x) - g(x)|$. I have seen how does the "other direction" work, I mean $(C[0,1],d_\infty \rightarrow (C[0,1],d_1)$ and I also ...
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Some question about $\ell_p$ and metric $d_p$ with $p\in (0,1)$

Hi I trying to see if $d_p=\sum_{k=1}^\infty|x_k-y_k|^p$ with $p\in(0,1)$ define a metric on $\ell_p$ my idea 1.first I need to see $\ell_p$ with $p\in (0,1)$ is not empty and $d_p$ Is good define. So ...
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Let $A\subset\Bbb R^m$ and $B\subset\Bbb R^n$ be closed subsets. Then $A \times B$ is a closed subset of $\Bbb R^{m+n}$

I am preparing for my exam right now and therefore practicing by doing some exercises. I need help for the following task: Let $A\subset\Bbb R^m$ and $B\subset\Bbb R^n$ be closed subsets. Then $A \...
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Smooth transition from Euclidean plane to hyperbolic plane

If I have a Poisson point process $\mathcal{X}$ of density $\lambda$ on the Euclidean plane $\mathbb{R}^2$, with the Euclidean metric taking pairs of points to the Euclidean distance, $$ \operatorname{...
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Convergence of measures on a compact metric space

In the paper "Ergodic optimization" by Oliver Jenkinson, Proposition 2.1 says (among other things): Let $T:X\to X$ be a continuous map on a compact metric space. If $f:X\to\mathbb{R}$ is ...
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$cl(A) \cap cl(B) = \emptyset$, then proof that $bd(A \cup B) = bd(A) \cup bd(B).$ [closed]

Let $A$ and $B$ are two subsets of a metric space $(X,d)$. Suppose that $cl(A) \cap cl(B) = \emptyset$, then proof that $bd(A \cup B) = bd(A) \cup bd(B).$ Where $cl(A)$ is the closure of the set $A$ ...
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If $c_n \nearrow c$ then $\lim_n \inf_{\pi \in \Pi(\mu, \nu)} \int c_n d\pi = \inf_{\pi \in \Pi(\mu, \nu)} \int c d\pi$

Disclaimer This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X,Y$ be Polish spaces and $c:X \...
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Show that if $(X,p)$ is complete, then so is $(X,d)$ as follows.

Let $d$ and $p$ be two metrics on $X$ such that $$\frac{1}{2022}p(x,y) \le d(x,y) \le \frac{p(x,y)}{p(x,y)+1},$$ for all $x,y \in X$. Show that if $(X,p)$ is complete, then so is $(X,d)$. Attempt: Let ...
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2 votes
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Non-linearity of Einstein's field equations

How to show that, for two Schwarzschild- metrics, the Ricci tensors of two metric tensors do not sum up linearly: $R_{\mu\nu} (g_1+g_2) \neq R_{\mu\nu} (g_1)+R_{\mu\nu} (g_2)$ while the Ricci tensor ...
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is there a NURBS but have weights on different axis?

We know that NURBS has the form $p(u)=\frac{\sum_{i=0}^n\ w_i\ N_{i,k}\ (u)\bf d_i}{\sum_{i=0}^n\ w_i\ N_{i,k}\ (u)}$ which adds weights on different control point $\bf d_i$. I am looking for a form ...
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Proof continuity of a function in a point by sequences

In my analysis class I learned there are at least three equivalent ways to look at continuity. Consider $f:A \subseteq\mathbb{R} \rightarrow \mathbb{R}$, then we can say: \begin{equation} \bullet \...
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Proving $|d(x,z)-d(y,z)| \leq d(x,y)$ and $|d(x,y)-d(a,b)| \leq d(x,a) +d(y,b)$ in a metric space $(X,d)$.

Let $(X,d)$ be a metric space, I want to prove the following inequalities: $$\tag{1}|d(x,z)-d(y,z)| \leq d(x,y),$$ and $$\tag{2}|d(x,y)-d(a,b)| \leq d(x,a) +d(y,b).$$ I understand $(1)$ as one side of ...
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The tent map system is transitive. Can we actually identify any of the infinitely many transitive points, however?

$\newcommand{\O}{\mathcal{O}}\newcommand{\G}{\mathcal{G}}\newcommand{\T}{\mathcal{T}}$TLDR; skip to the end of the preamble - we know that the tent map system is topologically transitive. However, do ...
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Given the given metric $ d(x_n,y_n)$, is the convergence of a sequence equivalent to pointwise convergence? [duplicate]

Let $X$ be the set of all complex sequences and let $ d: X \times X \rightarrow \mathbb{R} $ be defined by $$ d\left(\left(x_{n}\right)_{n \in \mathbb{N}},\left(y_{n}\right)_{n \in \mathbb{N}}\right)=\...
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Principal Argument not continuous using standard metrics

Let $Arg : C − \{0\} → R$ be the principal value of the argument, taking values in $(−\pi, \pi]$. Using the standard metrics on $C − \{0\}$ and $R$, show that $Arg$ is not continuous. I am thinking ...
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Realizing a neighborhood in $\mathbf{Gr}(k,n)$ as a neighborhood in $\mathbf{Gr}(1,n)$

Fix two positive integers $k,n$ with $1 < k < n$ and equip the Grassmannian $\mathbf{Gr}(k,n)$ with your favorite metric $\rho$. Now fix $\delta > 0$ and consider an open ball $B_{\delta}(x)$ ...
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Show that there exists open sets $P$ and $Q$ in $X$ such that $p \in P, S \subseteq Q$, and $P\cap Q = \emptyset$ in the metric space $(X,d)$

Let $(X,d)$ be a metric space, $S\subseteq X$ is finite, and $p \in X$, but $p \notin S$. Show that there exists open sets $P$ and $Q$ in $X$ such that $p \in P, S \subseteq Q$, and $P \cap Q = \...
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What are some examples of "Jordan spaces" which are *not* homeomorphic to a subspace of $\Bbb R^n$ (with the Euclidean topology)?

Note: This question has been substantially revised, see the edit history for earlier versions. So far, the only examples of metric spaces which I have seen in topology books are Euclidean $n$-space, ...
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What's the relation between (strict) convexity of unit balls and shortest distance paths in $l_p$ metric?

I'm reading the book Geometry of Quantum States by Bengtsson and Zyczkowski. They have a brief discussion on $l_p$ norms. Depending on circumstances, different choices of p may be particularly ...
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Let $A$ be a subset of a metric space $(X,d)$… [closed]

(c). Let $A$ be a subset of a metric space $(X,d)$. Prove that (i). $A$ is open if and only if $A=int(A)$. $$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [4\text{ marks}]$$ (ii). $A$ ...
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How to show that $A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}$ is closed?

I am having trouble with the following exercise: For $t > 0$ consider the set $$A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}.$$ ...
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Is the following metric space complete?

Let $X=\{x=(x_i)_{i \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}} \ \vert \ \exists N \in \mathbb{N} : x_i \geq 0 \ \ \forall i \geq N\}$ and let $\bar{\rho}$ be the uniform metric on $\mathbb{R}^{\...
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Two metrics $d_1$ and $d_2$ on $X$ are equivalent iff for each $A \subseteq X$, $d_1(x,A) = 0 \iff d_2(x,A) = 0 (x\in X)$.

I did the forward direction of this question. And my proof looks like this: Suppose $d_1(x,A) = 0$, where $x\in X$. So every $d_1$neighborhood of $x$ intersects $A$. That is $B_{d_1}(x,r)\cap A$ $\neq$...
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Arbitrary union of closed balls with constant radius

Intuitively, it feels like $\overline \bigcup_x B_\delta(x)= \bigcup_x \overline{B}_\delta(x)$ is true. $\overline \bigcup_x B_\delta(x) \supseteq \bigcup_x \overline{B}_\delta(x)$ is true for any set ...
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How to prove that every Cauchy net in a complete metric space is convergent? [duplicate]

How to prove the basic fact that every Cauchy net in a complete metric space is convergent? My definition: A metric space $X$ is complete iff every Cauchy sequence in $X$ is convergent. I don't ...
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2 votes
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Show that the discrete metric space $(\Bbb R^2,d)$ is complete.

Show that the discrete metric space $(\Bbb R^2,d)$ is complete. attempt: Let $(x_n)$ be an arbitrary Cauchy sequence in $(\Bbb R^2,d)$. We want to show that $(x_n)$ is convergent in $(\Bbb R^2,d)$. ...
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1 answer
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Locally compact Polish space is $\sigma$-compact?

I have recently encountered this result. Let $X$ be $\sigma$-compact, locally compact Hausdorff space and $\mu$ is a Radon measure on $X$. Then the space of continuous functions with compact support ...
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3 votes
1 answer
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The space of Lipschitz continuous functions is dense in that of uniformly continuous functions?

Let $(X,d)$ be a metric space. Then The space of bounded uniformly continuous functions is dense in that of bounded continuous functions w.r.t. the supremum norm. ref The space of bounded Lipschitz ...
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4 votes
1 answer
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Let $g(z)=1+e^z+e^{\alpha z}$ and $A=\{\Re(z) \mid g(z)=0\}$. Prove that $\overline{A}=[a,b]$.

Let $g:\mathbb{C} \to \mathbb{C}$ and $\alpha\in \mathbb{R}\setminus \mathbb{Q}$ such that $g(z)=1+e^z+e^{\alpha z}$ and let $A=\{\Re(z) \mid g(z)=0\}$.(Note that $\Re(z)$ is the real part of a ...
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Open set in pseudo metric space graphs

Good morning I have to find examples of open set in pseudo metric space. Let’s say: Let (X,d) be a pseudo metric space. Empty set is open in space (X,d). X is open in space (X,d) I’m looking for ...
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Riemannian distance via the length of a curve

I am reading a book where they define the Riemannian distance between two points on a manifold. Naturally it is given as the infimum of the integral of the length of the curves which connect the two ...
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Nonexistence of metric space on $\mathbb{R}$ in which only $\emptyset$ and $\mathbb{R}$ are open.

I'm trying to work through Analysis 2 script and prove a statement I came across. There exists no metric $(\mathbb{R}, d)$ in which only $\mathbb{R}$ and $\emptyset$ are open sets. (the script is in ...
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A (sub)neighborhood $S'$ of a set $X^*$ such that the line segment connecting any point in $S'$ and its projection to $X^*$ is contained in $S'$.

Consider a closed set $X^* \subset \mathbb{R}^n$. Let $Proj_{X^*}(x)$ denote the set of metric projections of $x \in \mathbb{R}^d$ to $X^*$: $$Proj_{X^*}(x) = \arg\min_{x^* \in X^*} d(x, x^*)$$ where ...
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3 answers
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Prove $d(f,g)= \min {|f(x)-g(x)}|$ in $C[a,b]$ is not metric

Trying to answer whether we can set in the set of real continuous functions in the closed interval $[a,b]$ metric $d(f,g)$ I found this For the min case, let $a=−1$ and $b=1$ and consider the ...
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1 answer
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The smallest compact set that contains a bounded set in a metric space

I came across the following assertion: Let $X$ be a metric space. Let $Y\subset X$ be a bounded set. Then, there exists a smallest compact set $K$ in $X$ such that $K\supset Y$. How can I prove this? ...
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1 vote
1 answer
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Can we guarantee the completness of a closed subset of a complete metric space?

As a second question: can we guarantee the completness of (the subset as subspace) a open subset $L$ of a complete metric space $M$ ? Complete subspace $L$, of metric space $M$, is a closed set in $...
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4 votes
1 answer
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Is the interval $[a,b]$ in $\Bbb{R}$ compact with the discrete metric?

I'm wondering whether or not a closed, bounded interval $[a,b]$ in $\Bbb{R}$ is compact with the discrete metric? I tried to show it wasn't by taking the set of singletons in $[a,b]$ as an open cover,...
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$\bar{B}_{1}(0) \subset \ell^{\infty}$ is not compact. [duplicate]

Show that: $\bar{B}_{1}(0) = \{x \in X | d(x,0) \leq 1 \} \subset \left\{\left(x_{k}\right)_{k \in \mathbb{N}} \subset \mathbb{R}^{n} \mid\left\|\left(x_{k}\right)_{k \in \mathbb{N}}\right\|_{\infty}&...
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Show that $d(\overline{x},\overline{y}) = 2^{-m}$ defines a metric.

Show that in the space of one-sided sequences, $X=\{\overline{x}= (x_n)_{n \in \mathbb{N}}\ ; x_n \in \mathbb{R} \}$, the function $$d(\overline{x},\overline{y}) = 2^{-m}$$ where $m$ is the biggest ...
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1 vote
1 answer
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Example of topologies generated by metric such that the union is not a topology?

I was going through the following exercise : $\cup_{\alpha \in \Lambda}\tau_{\alpha}$ is not a topology. $\tau_1=\{\{\phi\},\{a,b\},\{c\},\{a,b,c\}\}$ $\tau_2=\{\{\phi\},\{a\},\{a,b,c\}\}$ Let $U_1=\{...
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2 votes
1 answer
50 views

Close functions have close points

Suposse that $f,g:\mathbb{R}\to\mathbb{R}$ are $C^1$ and $$\int_0^1{|f-g|^2+|f'-g'|^2~dx}<\varepsilon$$ I would like to know if it is possible to say something about the distance between $f(x_0)$ ...
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Do the curvature properties of exotic spheres result in necessary new techniques for calculating arc length?

It was made clear to me in the post What answer does one get from integrating a Riemannian metric on a sphere for a great circle – does the metric’s non-flatness affect the answer? that a great circle ...
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Portmanteau theorem with lower semi-continuous and bounded from below functions

I'm trying to prove this version of Portmanteau theorem. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. ...
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Approximate a lower semi-continuous and bounded from below function by an increasing sequence $(f_n)$ of $n$-Lipschitz continuous functions

I'm trying to prove this result. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space. Let $f:X \to \mathbb R \cup \{+\infty\}$ be a lower semi-continuous and bounded from below. ...
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1 vote
0 answers
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$d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$.

I'd like some assistance in demonstrating $d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$ where $\mathbb{H}$ is a Poincare Upper Half-Plane, and $(pq;rs)$ is a cross ratio of $p,q,r,s$. ($p,q$ are on a geodesic ...
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2 votes
1 answer
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How is this version of Portmanteau theorem well-defined?

Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. I'm reading below theorem in this lecture note. Lemma 6.2. Suppose $\mu, \mu_1, \mu_2,\ldots \in ...
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0 votes
1 answer
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The product of metric topologies coincides with the topology induced by $p$-product metric

I'm trying to prove this interesting result. Could you verify if my attempt is fine? Let $(X_i, d_i)_{i=1}^n$ be a finite collection of metric spaces. Let $p \in [1, \infty]$ and $\| \cdot \|_p$ be ...
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  • 1,155
0 votes
1 answer
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How to show that the set $A=[a,b)$ is not open nor closed? [duplicate]

I only have the definition that a set $B$ is open if for all $x \in B$, there exists $\epsilon > 0 $ such that there is an open ball centered at $x$ and radius $\epsilon$ which is a subset of $B$. ...
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0 votes
2 answers
28 views

If $X$ is complete separable, then the space $\mathcal{P}(X)$ of all Borel probability measures on $X$ is separable in Prokhorov metric

I'm trying to prove below result. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal{P} :=\mathcal{P}(X)$ the space of all Borel probability measures on $X$. Let $d_P$...
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