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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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Showing that $M := \{f \in C([-1,1]) : f(0) > 0\}$ is open

Given $f \in C([-1,1],\mathbb{R})$ equipped with sup norm metric. I am trying to find out whether the subset $$M := \{f \in C([-1,1]) : f(0) > 0\} $$ is open/closed in $C([-1,1])$ My work: $M$ is ...
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How to determine whether the set is open or closed?

Let $f : \mathbb{R}^n\to \mathbb{R}$ continuous, and $a ∈ \mathbb{R}$. (1)$\{x ∈ \mathbb{R}^n : f(x) > a\}$ (2)$\{x ∈ \mathbb{R}^n : f(x) \geq a\}$ (3)$\{x ∈ \mathbb{R}^n : f(x) = a\}$ I don’...
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0answers
14 views

Convergence of null vectors [on hold]

Suppose we are given a sequence of null vectors $\{v_i \}$ in 3-dimensional Minkowski vector space. Is there a notion of convergence for the vectors, using the topology induced by the indefinite ...
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0answers
25 views

$ X \subset R^{2}$ be the subset $X=\{(x,y)|x=0,|y|\leq 1\} \cup \{(x,y)|0<x\leq 1,y=\sin(\frac{1}{x})\}$ is

Let$ X \subset R^{2}$ be the subset $X=\{(x,y)|x=0,|y|\leq 1\} \cup \{(x,y)|0<x\leq 1,y=\sin(\frac{1}{x})\}$ Then , X is a) compact b) connected c) path connected set. Graphically ,I have ...
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2answers
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proving that the space of sequences $M\ni x=(x_j\ :\ j\in\Bbb N)\subset A$, where $A$ is a set, with a certain distance is a complete metric space

Consider the space $(M,d)$ of sequences $x=(x_j :\ j\in\Bbb N),\ x_j\in A~\forall j$, where $A$ is a set, with $$d(x,y)=\begin{cases} \frac{1}{\min\{j\in\Bbb N^*\ :\ x_j\ne y_j\}} &\text{if}~~~ ...
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2answers
36 views

What are some advanced books on metric space?

Metric space, with the additional notion of “distance between points”, has properties that are more “concrete” than a topological structure. After a basic study I saw a number of strange and ...
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0answers
41 views

Countable open cover up to a null set

Given a metric space $(X,d)$,a probability measure $\mu$ (on the Borel sigma algebra) and an open cover $C:=\{A_i\}_{i\in I}$ of $X$, is it always possible to find a countable subset of $C$ that ...
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0answers
11 views

Explains to the distance of one closed and one compact set are alwasy greater than zero [duplicate]

One follow up question to this question Example to show the distance between two closed sets can be 0 even if the two sets are disjoint In this question, we get an example for A and B such that d(A,B)...
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1answer
35 views

Is it true that every totally bounded set in a metric space is compact?

Every compact set is totally bounded, but can we say that every totally bounded set is compact? I'm a beginner in metric space. My thinking is that a totally bounded set behaves like a finite set and ...
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2answers
48 views

If $int\overline{X} = int\overline{Y} = \emptyset$ then $int(\overline{X\cup Y}) = \emptyset$.

Let $X$ and $Y$ be subsets of a metric space. If $int\overline{X} = int\overline{Y} = \emptyset$ then $int(\overline{X\cup Y}) = \emptyset$. I know that $int(\overline{X\cup Y}) = int(\overline{X}\...
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1answer
35 views
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0answers
16 views

Wasserstein distance computation [on hold]

I have $2$ multidimensional series in $D (D>1)$ dimension. I am interested to compute Wasserstein distance between them to identify their similarity. Can anyone explain an easy way to compute it ...
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16 views

Open and closed subgroups of continous functions with max metric

Looking at the metric space C[-1,1] with the max metric. Show that only one of the following subgroups is open. How many of them are closed? $A = \{f \in C[-1,1] : f(x)<1 \,\,\,\forall x \in [-1,...
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1answer
18 views

How to find a completion for a metric space (For instance, support compact continuous real functions)

A completion of a metric space $(M,d)$ is a complete metric space $(M^*,d^*)$ such that $(M,d)$ is a dense subspace of $(M^*,d^*)$. I understand this, but how do I explicitely find a completion of a ...
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1answer
24 views

Proving that the non-negative real line is complete [duplicate]

Heads up: I am very new to abstract algebra and proofs. Take the non-negative real line $X = 0 \cup \mathbb{R}^+=[0, +\infty)$. We know that $X\subset \mathbb{R}$ and that $\mathbb{R}$ is complete. ...
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4answers
47 views

Find a metric $d$ on $\mathbb{N}$

Find a metric $d$ on $\mathbb{N}$ such that for any $n \in \mathbb{N}$ and any $\epsilon >0$ there exists an $m \in \mathbb{N},m\neq n$ such that $d(n,m)<\epsilon$. From this definition, all ...
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1answer
25 views

When does it make sense to find a point between two points?

I have this need to be able to express when a point is "between" two other points. One great example is the binary average operation $Avg:R \times R \rightarrow R$ that takes two real numbers and ...
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1answer
20 views

Prove that $f \in R(F_s)$ on the interval $[a, b]$ and that $\int f dF_s = f(s)$

Let $a<s<b$ and let $f:[a, b] \to \mathbb{R}$ be a bounded function that is continous at the point s. Define $F_s(x) = \begin{cases} 0, & \text{if $a \le x \lt s$} \\ 1, & \text{if $s \...
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1answer
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Is there a name for the function $D(x,y) = \sum\limits_{i = 1}^n (x_i - y_i)$, where $x$ and $y$ are vectors in $\mathbb{R}^n$?

I am studying real analysis and encountered this function $$D(x,y) = \sum\limits_{i = 1}^n (x_i - y_i)$$ where $x,y$ are $n$-dimensional vectors living in $\mathbb{R}^n$. It is easy to show that is ...
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3answers
33 views

Every inner product space is a metric space.

Show that every inner product space is a metric space. To show this should I set the distance metric as $d(x,y) = <x-y,x-y>$, then show properties of being metric space such as d(x,y) = d(y,x) ...
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0answers
11 views

Let $(X, d)$ be a metric space and let X be a finite set. Prove that every subset of X is open. [duplicate]

Let $(X, d)$ be a metric space and let X be a finite set. Prove that every subset of X is open.
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1answer
31 views

Does the intersection of two dense, open sets have the Baire property?

Definition: $A \subseteq X$ for a metric space $X$ has the Baire Property if for any sequence of sets {$V_{n}$} for $n \geq 1$ that are dense and open in $A$, $$cl(\cap V_{n}) \cap A = A $$ for all $...
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2answers
45 views

Prove that If B is open, then $\overline{A} \cap B \subset \overline{A \cap B}$

Let $(X,d)$ be a metric space and let $A, B \subset X$. Prove that: If B is open, then $$\overline{A} \cap B \subset \overline{A \cap B}$$ where $\overline{S}$ indicates closure for some set $S$. For ...
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0answers
56 views

Properties of a sequence $(s_n)$ such that $d(s_n,s_{n+1}) \rightarrow 0$.

I have been reading George M. Bergman's supplementary notes and exercises to Rudin's Introduction to mathematical analysis (which can be found here https://math.berkeley.edu/~gbergman/ug.hndts/...
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1answer
23 views

Global existence of IVP on $\mathbb R$

Suppose $f(t, x)\in C^1(\mathbb R^2)$ and satisfies $|f(t, x) |\leq 1+|x|$ for any $(t, x) \in\mathbb R^2$. Then prove that the IVP $x'(t) =f(t, x)$ and $x(0)=0$ has a solution defined on whole $\...
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2answers
50 views

Continuous function on a compact metric space with no fixed point

Let $(X,d)$ be a compact metric space, and suppose $f:X \to X$ is a continuous map with no fixed point. Show that there is an $\epsilon > 0$ so that $d(f(x),x) \ge \epsilon \ \forall x \in X$. I ...
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1answer
40 views

Show (F$_{k}$) converges uniformly to some continuous function

Suppose ${0<r<1}$. For each k $\in$ $\mathbb{N}$, define F$_{k}$ $\in$ C$\bigl($[-r,r]$\bigr)$ by F$_{k}$(x) = $\sum_{n=1}^k$ x$^{n}$. Show (F$_{k}$) converges uniformly to some continuous ...
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0answers
9 views

The size of a set with minimum interpoint distance in metric space

Given a metric space, $(X,d)$ with finite doubling dimension $ddim(X)$ (Namely, for every $r>0$, every ball of radius $r$ can be covered by $2^{ddim(X)}$ balls of radius $\frac{r}{2}$ ), and a set $...
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2answers
49 views

A one-to-one map between $M_{n\times n}$ and the $\mathbb{R}^{n^{2}}$?

I've been trying to think in something that can make this happen, but i'm not get anywhere. Plus, i have to show something through this map that can make this space($M_{n\times n}$) a metric space, so ...
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1answer
10 views

A closed set is always a subset of ball with arbitrary center and radius $r>2 \cdot diam(A)$

Let $(X,d)$ be a metric space and $A \subseteq X$. Let $B(x,r)$ be the open ball with center $x \in X$ and radius $r>0$. The set $A$ is closed, $diam(A)<r$ and also $A \cap B(x,r) \neq \...
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1answer
24 views

Let $Y$ be a complete metric space. Then $C^0 (X,Y)$ is complete under the uniform convergence metric.

I'm trying to show that the set of continuous functions $f: X \to Y$ is complete under the uniform convergence metric if $Y$ is complete. Just to be clear, the metric is: $$d(f,g) = \text{sup}\, \{e(...
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1answer
29 views

$X=\{1,2,3\},U=\{∅,\{1\},\{1,2\},\{1,2,3\}\}$ Is $(X,U)$ Hausdorff space?

Consider $X=\{1,2,3\}$ and a topology $$U=\{∅,\{1\},\{1,2\},\{1,2,3\}\}$$ Is $(X,U)$ a Hausdorff space? My solution: Points $x$ and $y$ in a topological space $X$ can be separated by ...
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2answers
40 views

If A complement is the union of two separated sets, prove that the union of those separated sets with A is connected.

Let $A$ be a connected subset of a connected metric space $(X,d)$. Assume $A^{c}$ is the union of two separated sets $B$ and $C$. Prove that $A \cup B$ and $A \cup C$ are connected. Attempt ...
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0answers
33 views

Particularize the Collage Theorem in $\mathbb{R}$

Consider the collage theorem stated as below (Interpolation and Approximation with Splines and Fractals by P. Massopust): I want to get the particular case for $\mathbb{R}$ with its usual distance, ...
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2answers
32 views

A concern on the definition of compactness in a metric space [duplicate]

Let $(X,d)$ be a metric space. This space is compact if any sequence $x_n \subset X$ has a convergent subsequence. This is how I'm given the definition of a compact metric space and it confuses me. ...
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3answers
42 views

Definition of the Lebesgue number of a open cover

Let $\mathcal{U}$ be an oper cover of a topological space $A \subseteq \mathbb{R}^n$. The Lebesgue number of $\mathcal{U}$ is defined as the least upper bound for all numbers $\delta \geq 0$ such that ...
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1answer
25 views

How to show that a given set is an open set?

$f,g:[0,1]\to \mathbb R$ continuous function and $f(t)<g(t), \forall t\in [0,1]$ $$U:=\{h\in C[0,1]:\forall t \in [0,1]:f(t) <h(t) <g(t) \}$$ in the space $X:C[0,1],||||_{\infty}$ I ...
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0answers
22 views

How to represent a conformal transformation using spinors?

In my mathematical meanderings, I've come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan ...
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1answer
346 views

Is the Set of Continuous Functions that are the Sum of Even and Odd Functions Meager?

Consider $X = \mathcal{C}([−1,1])$ with the usual norm $\|f\|_{\infty} = \sup_{t\in [−1,1]}|f(t)|.$ Define $$\mathcal{A}_{+}=\{ f \in X : f(t)=f(−t) \space \forall t\in [−1,1]\},$$ $$\mathcal{A}_{−}=\...
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1answer
38 views

$N$ should not be complete!!

The sequence ($\frac{1}{n}$) is a Cauchy sequence in $N$ but does not converge in $N$ as it converges to $0$.Then $N$ should not be complete. But $N$ is the closed subset of $R$ which is a complete ...
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2answers
31 views

Prove that $G$ is an open interval for two-valued continuous function $f$

Suppose $G\subset \mathbb R$ is a non-empty open set such that the function $f:G \rightarrow \{0,1\}$ is a two-valued function and is continuous. Show that any two-valued function on $G$ is a constant ...
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1answer
35 views

Proof $d(A \cup B,C) = \max\{d(A,C),d(B,C)\}$ in any complete metric space

I need to show: Let $(X,d)$ be a complete metric space. Show that if $A,B,C \in H(X)$ then $d(A \cup B,C) = \max\{d(A,C),d(B,C)\}$ (taken from Fractals Everywhere) Here $H(X)$ is the set of ...
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1answer
50 views

How do max and union commute in Hausdorff measure?

Recall that $d_H(A,B) = \max\{\max_{a \in A} \min_{b \in B} d(a,b),\max_{b \in B} \min_{a \in A} d(a,b)\}$ Theorem: Let $A,B,C \in H(X)$ (where $H(X)$ is the set of non-empty compact subsets of $X$)...
3
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4answers
62 views

Is the set $\{ (x,y) \in \mathbb{R}^2 : xy=1 \}$ open or closed in $\mathbb{R}^2$

Determine whether the following sets are open or closed in $\mathbb{R}^2$ endowed with the eucledian metric $1. \{ (x,y) \in \mathbb{R}^2 : xy=1 \}$ $2. \{ (x,y) \in \mathbb{R}^2 : xy\le1 \...
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1answer
34 views

Level curves of $f(r) = \sum_{i=1}^na_id(r,r_i)$ with $r_i\in \mathbb{R}^n$

Let $$f(r) = \sum_{i=1}^na_id(r,r_i)$$ Where each $r_i \in \mathbb{R}^n$ and each $a_i\in \mathbb{R}$ and $d:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ denotes the usual distance function. ...
0
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1answer
23 views

Are euclidean and rectilinear metrics equivalent for determining the ordering of closeness between points?

I'm writing software that necessitates calculating hundreds of thousands of distances between points (in this case, in 67-space). The distance between two points $p$ and $q$ using euclidean metrics ...
0
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2answers
27 views

Show $(\sum_{k=1}^{n} |u_k|^2)^{\frac12}\leq \sum_{k=1}^{n}|u_k| \leq n\cdot max\{|u_k|:1\leq k \leq n\} $

My text book has used the following inequalities without any proof: Let $X=R^n$, then $$(\sum_{k=1}^{n} |u_k|^2)^{\frac12}\leq \sum_{k=1}^{n}|u_k| \leq n\cdot max\{|u_k|:1\leq k \leq n\} \leq n\...
0
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1answer
39 views

A set F is closed if and only if every convergent sequence in F converges to a limit in F

$F \subseteq X$, where $X$ is a metric space. Then $F$ is closed if and only if every convergent sequence in F converges to a limit in $F$. Attempt $ \implies$ If $F$ is closed, $F= \overline F$, ...
0
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1answer
25 views

Show sup(A) is in F, where F is closed and open

This question is worded very strangely and honestly I'm rather confused. Suppose $\varnothing$ $\subsetneq$ F $\subsetneq$ $\mathbb{R}$ is closed. Then $\exists$ x$_{0}$ $\in$ F and y$_{0}$ $\in$ F$^{...
1
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1answer
28 views

Metric on compact set with one point less

Let $K$ be a compact metric space. Now remove one point from it. Is there a metric on $K$ with one point removed that generates the same topology as the initial metric but is such that the space $K$ ...