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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Proof that the nowhere differentiable functions are dense in $C_b(\mathbb R)$.

I tried to make a proof, where I use a Weierstrass function. I was surprised at how easy it was, and thus a little doubtful as to the correctness of the proof. I've looked it over, and didn't find any ...
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Proof idea: Let $(X,d)$ be a metric space, and $\rho$ be bounded metric, show that they will generate the same topology

Let $(X,d)$ be a metric space, $d$ generates the metric topology $\mathcal{T}$ via metric ball $B_\epsilon(x)$. Show that bounded metrics: $\rho_1(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ with ...
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Show that $d_1=\min(d(x,y),2)$ is a metric space [duplicate]

Show that $d_1=\min(d(x,y),2)$ is a metric space if it is given that $d(x,y)$ is a metric space. I am stuck at the triangle inequality part, to show that $d_1(x,z)\leqslant d_1(x,y)+d_1(y,z)$ i.e ...
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58 views

Brute force way to show that $\rho(x,y) = \min\{1, d(x,y)\}$ is a metric

Following a hint in Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric I would like to use the brute force method to show that the standard bounded metric is a metric $$\rho(x,y) ...
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Show compactness of subset of $\mathbb R^3$

I need to show that $$A:=\{(x,y,z)\in\mathbb R^3; 3x^3y+2xyz^3+2y^2+3=0, xy^3+3xz+x^3=0\}$$ is closed and bounded, hence compact. I don't really know what to do here, can you help?
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Follow up to a question, why does proof $\rho(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ work

This is a follow up to a well known question Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric A general proof is as follows: Let $x,y,z \in (X, \rho)$ \begin{align} \rho(x,z) &= \dfrac{...
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Any metric space can be isometrically embed in some Banach space? [duplicate]

I have just read the question of the title in an article from Kirchheim. I didn't know this result, does any one know where I can find a proof of it?
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A complete metric on $\mathbb{R}$

Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a function. What conditions must $\varphi$ satisfy so that the metric space $(\mathbb{R},d_\varphi)$, where $d_\varphi(x,y)=|\varphi(x)-\varphi(y)|$, is ...
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To find two points in compact metric space satisfying specific property

Let $(X,d)$ be a compact metric space.Suppose that for all positive real numbers $t<1$ ,there are points $x_{t}$,$y_{t}$ such that $d(x_{t},y_{t})=t$.Prove that there are points $x$, $y$ in $X$ ...
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metric space with no perfect set

Let $X$ be a complete separable metric space containing no perfect set of size greater than $1$. In other words every subset of $X$ has an isolated point. It is well known that $X$ must be countable....
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Is diameter of a set a measure?

Suppose the diameter of a nonempty set $A$ is defined as $$\sigma(A) := \sup_{x,y \in A} d(x,y)$$ where $d(x,y)$ is a metric. Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable ...
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Is there any metric $d$ on $\mathbb R$ and $a \in \mathbb R$ such that the function $f:\mathbb R \to \mathbb R$, $f(x)=d(x,a)$ is differentiable?

Let $d$ be any metric on $\mathbb R$ , then I know that the two variable scalar field $f: \mathbb R^2 \to \mathbb R$ , $g(x,y)=d(x,y)$ is never differentiable . Now what I want to ask is this : Let $a ...
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$X$ is metric space s.t. for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ ; is $X$ compact?

Let $X$ be a metric space such that for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ , then is $X$ compact ? Compare with this $A \subseteq \mathbb R^n $...
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$A \subseteq \mathbb R^n $ s.t. for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ , is $A$ closed $\mathbb R^n$?

Let $A \subseteq \mathbb R^n $ such that for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ ; then I know that $A$ is bounded ; my question is , is $A$ closed in $\...
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Why does one necessarily need the triangle inequality

I'm studying basic topological, metric and normed spaces and I am curious why one of the axioms of both a metric and a norm is the triangle inequality. It makes some sense to me having the triangle ...
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28 views

Difference between norm and distance. [duplicate]

I was wondering the difference between norm and distance. My teacher told me that a norm always induce a distance, but that the reciprocal is not true. So, let $(E,\|\cdot \|)$ a normed space. I agree ...
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34 views

Hints on showing that a metric space is complete

Let $C[0,K]$ be the space of all continuous real valued functions on $[0,K]$ for $K>0$ and $L\geq0$, equipped with the metric $d$ defined by $$d(f,g)=\sup_{0\leq k\leq K}e^{-Lk}|f(k)-g(k)|.$$ I ...
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1answer
36 views

Find all near points in a large array of points

Simplified problem: I have an array of points in 3D space. I want to find all pairs that are within a given distance from each other. (I'm writing a very simple simulation and the points merge into a ...
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1answer
164 views

Existence of a homeomorphism that does not return much

Let $f:X\rightarrow X$ a homeomorphism where $X$ is a compact metric space. Fix $x\in X$, denote $O(f,x)=\{ f^n(x):n\in \mathbb{Z}\}$ the orbit of $f$ by $x$. For $m\in \mathbb{N}$ denote $O(f,x,m)...
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Proof about a Topological space being arc connect

While reading a book i found a topological space described as: Let $(X,\tau)$ be the topological space formed by adding to the ordinary closed unit interval $[0,1]$ another right end point,say ...
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1answer
36 views

Continuous image of a cantor set and other space filling curves

I am studying the theory of space filling curves, more specifically looking at the continuous image of a Cantor set. I have been given this definition to characterise the continuous image of a Cantor ...
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1answer
31 views

Limit point - contains one, or infinitely many points of set?

I'm reading through Functional Analysis by Bachman. He defines a limit point as follows: The point $x$ is said to be a limit point of $A \subset X$ iff for every $r$, $S_r(x) \cap A$ contains ...
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Prove that $f(x) = \cos(x)$ has a unique fixed point.

The following question comes from Chapter 6.4, Exercise 4 on page 156 in the set of notes Topology Without Tears. Using Exercise 2 and 3, show that while $f: \mathbb R \to \mathbb R$ given by $f(x)...
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1answer
17 views

If $X$ and $Y$ are homeomorphic, then for every $A\subset X$, $X-A$ and $Y-f(A)$ are homeomorphic.

Let $X$ and $Y$ be metric spaces and $X$ and $Y$ are homeomorphic under $f:X\to Y$, then for every $A\subset X$, $X-A$ and $Y-f(A)$ are homeomorphic. It is quite intuitive but how can we write the ...
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Is this collection of function uniformly equicontinuous? Hints on the proof.

Let $f_n(x)=\frac{1}{n}\cos(e^{nx})$ for $n\in\mathbb{N}$ be a sequence of functions for $x\in[0,1]$. Is it true that $\{f_n\}$ is a uniformly equicontinuous collection of functions? My attempt so ...
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1answer
28 views

If $f:X\to Y$ is continuous and $X$ is a totally bounded metric space, is $f(X)$ also bounded?

If $X$ and $Y$ are metric spaces and $X$ is totally bounded and $f:X\to Y$ is continuous (not necessarily uniform), is it true that $f(X)$ is also bounded? How can we prove it or is there any counter ...
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36 views

Lines in a metric space - a metric space?

In a metric space, a point $x$ is between points $u$ and $v$ if $d(u,v)=d(u,x)+d(x,v)$. The line determined by points u and v consists of $u$, $v$ and all points $x$ such that one of $x,u,v$ is ...
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1answer
56 views

Net convergence vs sequence convergence

I am stuck trying to solve the following exercise regarding nets and sequences: Let $(x_n)_{n\in \mathbb N}$ a sequence in the metric vector space $(V,d)$. Let $\mathcal M$ be the set containing ...
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Show that if $f$ continuous then $f(\overline{X})\subset \overline{f(X)}$ for all $X\subset R$

I was able to prove that $f$ not continuous $\implies f(\overline{X})\not\subset \overline{f(X)}$ by doing this: Being $f$ not continuous in $a$, there exists a sequence $x_n$ with $\lim x_n = a$ ...
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How to show a set is compact using sequential compactness definition?

Let $l^{\infty}$ be the vector space of all bounded sequences $x=(x_n)$ of real numbers with the norm $||x||=\sup_{n\in\mathbb{N}}|x_n|$ and $l^{\infty}$ is complete. I am trying to show that the set ...
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Pseudometric without triangle inequality

I'm working on an optimization problem where I aim to minimize the total euclidean distance of the edges of a graph drawn on a fixed-size grid. For convenience, I actually use the maximum of $0$ and ...
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Proximity to a Shortest Path

So I'm trying to find Proximity of a point $x$ to a Shortest Path between $a$ and $b$. Things I think I know, given I have a metric $d(i,j)$ for the SP: There's no inner product (because there's no ...
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Bounds on Hausdorff distance via singular values

For some $\delta>0$, let $X$ and $X_\delta$ be two bounded convex polytopes in $\mathbb{R}^n$, defined as $X = \{x \in \mathbb{R}^n : Ax \leq b \}$ and $X_\delta = \{x \in \mathbb{R}^n : Ax \leq b +...
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1answer
34 views

Derivation of metric from product topology?

Suppose you have two topological spaces, g11 and g22, which are "components" of a more general topology. For example, suppose that a metric has components g11, g12, g21, and g22. And suppose you want ...
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Can $f(x,z) = x^Tx + \sum\limits_{i = 1}^n \dfrac{x_ix_i}{z_i}$ be written with multiple inner products at the same time?

I am running into a very interesting phenomenon that I do not quite understand (Illustration of an example of so called subset of $\mathbb{R}^n$) For example, suppose we have a subset of $X \...
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415 views

A topology on the set of lines?

Of course any set $X$ can have a topology, but are there more natural topologies, metrics or similar on the set of straight lines in $\mathbb R^2$?
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Counterexamples about locally compact sets on the real line

Is there a counterexample in the space $\mathbb{R}$ with it's usual metric to the statements: The union of two locally compact subsets of $\mathbb{R}$ is locally compact The complement of a locally ...
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What is the connection between positive definite hessian and metric?

In heard from a someone in verbatim that if you take the taylor series of a certain function, if the Hessian is positive definite, then it is a metric. This quote is without context and therefore ...
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Separable metric space with no isolated points.

Let $(X,d)$ be a separable metric space with no isolated points and $(X,\mathcal{B}(X),\mu )$ is a measure space such that $\mu(X)<\infty$. How to prove that for all $\epsilon >0$, there exists ...
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Prove that $ (X, d) $ is a metric space

Let $ X $ be the set of $ n$-letters word, that is $ X = \{(x_{1}, x_{2}, \dots, x_{n}) \} $ where $ x_{i} $ is an alphabetical character. Define $ d(x, y) $ between two words $ x = (x_{1}, \; x_{2}, \...
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$f(x)=0$ for $x\in X$, then $f(x) = 0$ for $x\in\overline{X}$

I'm trying to prove: $f(x)=0$ for $x\in X$, then $f(x) = 0$ for $x\in\overline{X}$ where $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ and is continuous. I know that the set $\overline{X}$ is ...
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Why is this equality relevant in this proof of closedness?

I've got a theorem:For any set $A$ in a metric space: $\left(\overline{A}\right)'=A'$ and the book proves a corollary: $\overline{A}$ is closed. The proof is this: $\left(\overline{A}\right)'=A'\...
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1answer
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$X$ open $\implies A = \{x\in X; f(x)\neq g(x)\}$ is open, and $X$ closed $\implies F = \{x\in X; f(x)= g(x)\}$ is open

I need to prove: $X$ open $\implies A = \{x\in X; f(x)\neq g(x)\}$ is open, and $X$ closed $\implies F = \{x\in X; f(x)= g(x)\}$ is open I've found this question that basically proves it for the ...
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75 views

Uncountable sets in metric spaces

Suppose we work in a separable, complete metric space $X$. Let $Z$ be an uncountable subset of $X$, must there exist $x_0\in Z$ and a sequence $(x_n)_{n=1}^\infty$ of elements in $Z$ different from $...
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2answers
90 views

If $A \subseteq B$, does $\mbox{dist}(x,\partial A) \le \mbox{dist}(x, \partial B)$ hold for all $x \in A$

Let $A,B \subseteq \mathbb{R}^n$ with euclidean metric. Furthermore let $$ \mbox{dist}(X, Y) := \inf\{|x - y| : x \in X, y \in Y\}. $$ Does the following implication hold? $A \subseteq B \implies \...
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51 views

An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open. [duplicate]

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
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48 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
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1answer
27 views

Base of the Baire space [closed]

Why base of the Baire space is countable? Because it is a perfect Polish space, which means it is a completely metrizable second countable space with no isolated points.
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17 views

Equivalence between properties of compactness for metric spaces

I am attempting here to show the equivalence between the following three statements for the metric space $(X,d),$ i) $(X,d)$ is compact, meaning every open cover admits a finite subcover ii) $(X,...
4
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2answers
39 views

Identical Geodesics implies scalar multiple of metric?

Suppose $(M,g^1)$ and $(M,g^2)$ are two intrinsic metric spaces with the same underlying set $M$. Assume that for every $p,q\in M$, for each geodesic $\gamma^1_{[p,q]}$ connecting $p$ to $q$ under $...