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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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Prove that if $\mathfrak A$ contains a bounded set then $\exists M \in \Bbb Z^+$ s.t $A_1 \cap \cdots A_M=∅$

Suppose $\mathfrak A$ is a set of closed subsets of $\Bbb R$ such that $∩_{A∈\mathfrak A}A = ∅$. Prove that if $\mathfrak A$ contains a bounded set then $\exists M \in \Bbb Z^+$ s.t $A_1 \cap \cdots ...
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How to prove inequality $\sum_{j=1}^n\vert x_jy_jz_j\vert\le\Bigl( \sum_{j=1}^n\vert x_j\vert^{p_1}\Bigr)^{\frac{1}{p_1}}…$

I'm new in StackExchange and in functional analysis. I have a problem with one exercise: If $p_1, p_2,p_3\gt1$ are real numbers with $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=1$ is true, n - a ...
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Subgroups of $\mathbb{R}^n$ isomorphic to integer lattices [duplicate]

Consider $(\mathbb{R}^n, +)$, endowed with standard metric. Let $H$ be its subgroup, with the property that there is a neighbourhood of $0$, $V$, such that $V \cap H = \{0\}$. How do I prove that $H \...
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Is the following subset of $\mathbb{R}^2$ open, closed or neither?

I am given the subset $A=\{(x, \sin(1/x)) | x>0\} \bigcup \{(0,y) | y \in [-1,1] \}\subset \mathbb{R}^2$ equipped with the standard Euclidean metric I came to the conclusion that it is not open ...
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Group action and invariant measures

Let's first start with the example from where this question arose. I consider the metric space $\mathbb{R}$ and the Lebesgue-measure $\lambda$ on $\mathbb{R}$ as well as the transitive group action $+\...
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Reachability of points on Manifolds

I was thinking with a friend that on a surface some points are more reachable than other.In the sense that their average distance to the other points is lower. e.g. suppose that we have a circle in a ...
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Do the partial sums of a divergent series converge to Cesaro or Abel sums in some metric?

Let $(a_n)$ be a sequence in $\mathbb{R}$, and let $s_n$ be the $n^{th}$ partial sum of the sequence. Then the Cesaro sum of $(a_n)$ is the limit of the average of the first $n$ partial sums as $n$ ...
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4answers
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Locally compact metric space having atleast one non-compact closed ball

Can we have a non-trivial example of a locally compact metric space in which atleast one non-trivial closed ball is not compact. I am considering that infinite set with discrete metric is a trivial ...
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Equidistant points in arbitrary metric

I am looking for a way to place $2^n$ points equidistantly on a space with low dimensions. Using Euler distance as a metric, for example, 4 points can be placed in three dimensional space, which forms ...
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Let $(X,d)$ be a metric space, let $x\in X$, let $\delta <\epsilon$. Then, $\overline{B(x, \delta)} \subset B(x,\epsilon)$.

Let $(X,d)$ be a metric space, let $x\in X$, and let $\delta <\epsilon$. Then, $\overline{B(x, \delta)} \subset B(x,\epsilon)$. My attempt:- Let $y\in \overline{B(x, \delta)}$ iff every open set ...
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Two real discrete sets with non discrete union

I need to find an example of two discrete sets $X,Y\cap \mathbb{R}$ such that $X\cup Y$ is not discrete. I would like to share my attempts, but I'm not seeing any way to think. I need some tips, I ...
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Let $S=\{(x,y) \in \mathbb{R}^2: 0<x^2+y^2 \leq 1\}$ be a subset of the metric space $M = \mathbb{R}^2$, prove that $\lim(S) = \{(0,0)\} \cup S$

Let $S=\{(x,y) \in \mathbb{R}^2: 0<x^2+y^2 \leq 1\}$ be a subset of the metric space $M = \mathbb{R}^2$, prove that $\lim(S) = \{(0,0)\} \cup S$ $\lim(S)$ is the set of limit points of $S$. By ...
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“Almost metric” functions

The conditions for $d:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ be a metric are: d1) $d(x,x)=0$ d2) If $x\neq y$, $d(x,y)>0$ d3) $d(x,y)=d(y,x)$ d4) $d(x,z)\leq d(x,y)+d(y,z)$ I'm ...
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Polynomial ring as a (complete) metric space

Let $k$ be a field of characteristic zero (for example $\mathbb{R}$ or $\mathbb{C}$), and let $R=k[x_1,\ldots,x_n]$ be the $k$-algebra of polynomials in $n$ variables $x_1,\ldots,x_n$, $n \geq 1$. ...
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Does the completeness of $\mathbb{R}$ mean the metric space $(\mathbb{R},d_E)$ over the field $\mathbb{R}$ complete?

As far as I know, completeness is a property of topological vector space whose topology induced by a metric. While $\mathbb{R}$ is a field, generally speaking, a set, rather than a space. But ...
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To show the two metrics induce the same topology but only one of them is complete

We have a standard metric $d_1(x,y) := |x-y|$ on $\mathbb{R}$, and another metric $d_2(x,y) := |f(x)-f(y)|$ on $\mathbb{R}$, where $f(x)=\frac{x}{1+|x|}$. I want to show these two metrics induce the ...
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Checking whether $X=\mathbb R$ with $d(x,y)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ is a metric space

Examine whether $d$ is a metric on $X=\mathbb{R}$ where $d\left(x,y\right)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ for all $x,y\in \mathbb{R}$ I think it is not. Even though it satisfies all other properties ...
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How could one geometrically visualize any given metric space $(X,d)$?

Example. Say $X=\mathbb{R}$ and $d(x,y)=\frac{d_0(x,y)}{1+d_0(x,y)}$ where $d_0(x,y)=|x-y|$ is the Euclidean metric. The visualization of e.g. $\mathbb{R}$ with Eucledian distance $d_0(x,y)=|x-y|$ is ...
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How is 0 a limit point of $\{1/n\}_{n=1}^{\infty}$?

I'm working through Principles of Topology and can't wrap my head around how $0$ is a limit point of $\{1/n\}_{n=1}^{\infty}$. For reference, this is on page 44 of the 2016 edition. The book provides ...
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Homework question on convergence using the p-adic metric

I am self teaching myself about metric spaces (officially a physics student), and have come across the following question: I am asked to show that the sequence 215, 2015, 20015, 200015... converges ...
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1answer
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Is empty set with usual metric space $(X=\mathbb{R},d)$ bounded?

Here R is set of real numbers and $d$ is distance metric i.e $|x-y|$. I wanted to find if empty set under given metric space is compact and we know by Heine Borel theorem that a set is compact iff it ...
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1answer
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$(\mathbb{Q},|\cdot|_{\mathbb{Q}})$ is not connected

Hello i have this exercise: let $(E,d)$ a connected metric space, with an unbounded metric, show that the sphere is not empty, after deduce that $(\mathbb{Q},|.|_{\mathbb{Q}})$ is not connected. I ...
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1answer
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Show that $\{ x = (x_n) \in H^\infty: |x_k| < 1, k = 1, …, N\}$ is open in $H^\infty$

Let $H^\infty$ be the hilbert cube, with metric $d(x,y)=\sum_{n=1}^{\infty}\frac{x_n-y_n}{2^n}$ Show that $\{ x = (x_n) \in H^\infty: |x_k| < 1, k = 1, …, N\}$ is open in $H^\infty$ I think you ...
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Changing the distance considered on a metric space changes open sets inside?

I've just started topology on my grade. It makes a lot of sense to start introducing something that we've already been introduced to before on Calculus I, the notion of metric spaces. My teacher said ...
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Open sets and open balls in a metric space

Theorem. Suppose $(X,d)$ be a metric space. If $U\subseteq X$ is an open set, then for all elements $a\in U$, there exists a real number $\delta> 0$ such that $B(a;r)\subseteq U$. I tried to ...
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1answer
35 views

To check whether given space is metrizable

Let $X$ be a compact Hausdorff topological space without isolated point . Also, there exist open cover $\mathcal{V}=\{V_i\}_{I=1}^m$ and a homeomorphism $f:X\to X$ such that $card(\bigcap _{j\in\...
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2answers
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Prove that *every* subset of a metric space $M$ can be written as the intersection of open sets. [duplicate]

Prove that every subset of a metric space $M$ can be written as the intersection of open sets. My attempt: If $A\subset M$ is open, $A$ can be written as $A\cap M$, which is the intersection of 2 ...
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How to tell if a set is bounded? And the difference between a ball and sphere?

I am given the definition of a ball vs sphere but I can't get an image in my head. I can see a circle for a dotted edge for a ball, and a circle with a 'lined' edge for a sphere. In $\mathbb{R}^n$ are ...
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Show that the set $F = \{ (x, y) \in \mathbb{C} \mid x \ge 0, y\ge 0 \}$ is a closed subset of the complex plane $\mathbb{C}$ [closed]

This is a question for my metric space assignment. Although, I am able to identify that we will have to assume that there are two balls on the axes, I am not sure how to present my answer and what to ...
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Prove that a subset in Hilbert space is open

I'm a undergraduate who enjoys math and follows his first course on real Analysis. I follow this course for fun and i'm stuck on the following problem: Let $H^{\infty}$ be the Hilbert cube (the ...
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2answers
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Closed subset of $\mathbb{R}^n$ and frontier

Let $F$ be a closed subset of $\mathbb{R}^n$. Show that there exists $X \subset \mathbb{R}^n$ such that $\partial X = F$ (Frontier of $X$ is equal to $F$). Is this fact is true in general, i.e., for ...
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Can a disk be defined with a discrete metric?

$D=\{(x,y)\in \mathbb{R^2}:\rho_d(x,y)<\epsilon\}$? Say for $\epsilon<1$, the disk doesn't exist. Edit: Elaborating on the question. Can you call it a disk if it is not the euclidean metric?
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If $A \subseteq X$ is open and $G \subseteq A$, then $G$ is open in $A$ iff $G$ is open in $X$

I'm having some trouble solving this problem: Show that if $A$ is an open set in $(X, d_X)$, then a subset $G$ of $A$ is open in $(A, d_A)$ if and only if it is open in $(X, d_X)$. It seems that ...
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2answers
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How to show a topological space is not metrizable?

Define the topological space by taking the Reals and identifying all numbers divisible by 10 to a single point, call this space $∆$. How do we show that $∆$ can not have a metric associated to it and ...
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2answers
52 views

Is every set metrisable?

Can any set be turned into a metric space? Let $X$ be any set, let $x,y\in X$ define a metric $d:X\times X\rightarrow \mathbb{R}$ by: $$ d(x,y) = \left\{\begin{array}{cc} 1 & x\neq y \\ 0 & x=...
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1answer
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Can the topology generated from the lexicographic order on $\mathbb{R}^2$ come from a metric?

Can the topology generated from the lexicographic order on $\mathbb{R}^2$ come from a metric? I'm assuming (not told otherwise) that the metric on $\mathbb{R}^2$ is the trivial one. My gut says that ...
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Covering number of metric space

For a separable metric space $(X,d)$, we say it satisfies the global $N$ covering property if any ball $B_r(a)$ can be covered by $N$ balls of radius $r/2$. Let's see an example, let $(\mathbb{R}^2,d)$...
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1answer
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Metrics that generate the same convergent sequence.

$(*)\ c_1 d_1(x,y) \le d_2(x,y) \le c_2 d_1(x,y)$ for some constants $c_1,c_2 > 0$. Let $d_1 = |x-y|$ and $d_2 = \sqrt{|x-y|}$ be metrics on $\mathbb{R}$. Show that these metrics do not satisfy $(*...
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If two spaces are homeomorphic and one is a metric space must the other be as well?

Suppose $X$, and $Y$ are topological spaces, and that they are homeomorphic. If $X$ is a metric space must $Y$ be as well? Here is my thought process. If $X$ is a metric space then there exists some ...
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1answer
26 views

There are no nontrivial metric spaces such that every sequence is a Cauchy sequence

Is my intuition correct here? If our metric space consists of a single point $x$, any sequence in it would just be $x, x, x, \dots$ which is obviously Cauchy. If we have any metric space $(M, d)$ ...
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Elements of infinite order in CAT(0) groups

In E. Swenson, A cut point theorem for CAT(0) groups, J. Differential Geom. 53 (1999), no. 2, 327–358. the author shows (Theorem 11) that if a group $G$ acts geometrically (i.e. properly ...
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1answer
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Purpose of Metrizability

What is an example of when metrizability is used to prove a result? Or is metrizability of a more philosophical nature?
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$x$ is a limit point of $A$ is and only if there exists a sequence of distinct elements which converges to it. [duplicate]

I have probably seen and done this problem a hundred times, but thinking back on it, I realized something which I hadn't before. Here, I am only concerned with the forward direction of the proof, and ...
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1answer
31 views

A sequence of functions decreasing to 0 is equicontinuous in a compact metric space.

How to proof that? Let M be a compact metric space and $ \{f_n\} \subset C(M,\mathbb{R})$, so that $\{f_n\}$ is decreasing and $ lim f_n(x)=0 $, then $\{f_n\}$ is equicontinuous
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2answers
113 views

Does there exist a continuous bijection from real line onto the unit circle? [duplicate]

Does there exist a continuous bijection from $\mathbb R$ onto $S^1$ ? I know that there isn't any continuous bijection from $S^1$ onto $\mathbb R$ because such a continuous bijection would be a ...
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2answers
871 views

A continuous bijection between two complete metric spaces that is not a homeomorphism.

Suppose $X$ and $Y$ are two metric spaces and $f: X\to Y$ be a continuous bijection. Now my question is does the completeness of $X$ and $Y$ implies $f$ to be a Homeomorphism? My idea. First of ...
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2answers
65 views

Characteristics of a convex set if its boundary is convex

If $A$ is a convex set in $\mathbb R^n$, when is its boundary convex as well? I think $\partial A$ must be either contained in a hypersurface or must equal $\mathbb R^n$.
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+50

Calculating the boundary of a specific subset of real numbers

Consider the subsets {$J_m| m=0,1,2,...$} of $\mathbb R$ defined by $$J_m:= \bigcup_{s=0}^{m-1} \bigcup_{n\in \mathbb Z} \left(\frac{3^m(2n+1)+2\times 3^{s}+1}{3^m},\frac{3^m(2n+1)+2\times 3^{s}+2}{3^...
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1answer
34 views

Show that $d(x+y,A+B) \leq d(x,A) + d(y,B).$

Problem: Let $(X,d)$ be a metric space. For each nonvoid subsets $A$ of $X$ let $$d(x,A) = \inf_{y \in A} d(x,y), \quad x \in X.$$ Show that: $d(x,A) = d(x,\bar{A})$. In particular, $d(...
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1answer
39 views

In a complete Riemannian manifold $M$, do any two points $p,q\in M$ admit a length minimizing geodesic?

Clearly, completeness gives us the existence of at least one geodesic $\gamma:[0,1]\to M$ such that $\gamma(0)=p$ and $\gamma(1)=q$, however, this geodesic need not be length minimizing. I strongly ...