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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$X$ 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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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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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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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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30 views

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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55 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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41 views

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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16 views

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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39 views

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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1answer
27 views

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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1answer
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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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33 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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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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37 views

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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20 views

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: $\left(\overline{A}\right)'=A'\...
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$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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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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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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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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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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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,...
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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 $...
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Prove that $(\mathcal{F}(E, \mathbb{R}), \|.\|_{\infty})$ is complete

Let $f_n$ be Cauchy for the $\|.\|_{\infty}$ norm, meaning we have $$\forall \varepsilon > 0, \exists N \in \mathbb{N}, \forall n \geq N, \forall p \in \mathbb{N}, \|f_n - f_{n+p}\|_{\infty} < \...
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The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
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How do I show that $d:\mathbb{R^2}\times\mathbb{R^2}\rightarrow\mathbb{R}$ is a metric defined in $\mathbb{R^2}$? [closed]

If $d(\vec{u},\vec{v}) = \lvert u_1-v_1\lvert+\lvert u_2-v_2\lvert$ for $\vec{u}=(u_1,u_2),\vec{v}=(v_1,v_2)$. How can I show that $d$ is defined in $\mathbb{R^2}$? Would it be enough to show the ...
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Question about proof of the tube lemma for metric spaces

Tube lemma: Let $M$ be a metric space and $K$ a compact metric space. Let $a\in M$, $a\times K\subset V\subset M\times K$, that is, suppose there is an open set $V$ between $a\times K$ and $M\times K$...
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Intersection of Compact sets Contained in Open Set

Just wanted to see if my proof of the following is valid: Let $\{K_i\}_{i=1}^{\infty}$ be compact sets (in some metric space), and let $V$ be an open set such that $$ \bigcap_{i=1}^{\infty} K_i \...
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1answer
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Computing Hausdorff metric for some sets

Just started to learn about metric spaces, and I came across the Hausdorff metric. Let $K$ be the family of non-empty closed subsets of $[0,1]$. For $A \in K$ and $\delta > 0$ let $A_{\delta}$ be ...
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if the metric $d_1$ is complete, and $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$, is $d_2$ complete?

two metrics $d_1, d_2$ on $X$, For all $x_n$ and $x$ from $X$ it holds : $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$ Is it true that $(X, d_1)$ complete implies that $...
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Metric Matrix of the hyperbolic Riemannian manifold

Let $\Bbb{H}^n:=\left\{(x_1,...,x_n)\in\Bbb{R}^n\mid x_n>0\right\}$ be the hyperbolic space and $g={d^2x_1+\dots+d^2x_n \over x_n^2}$ be the standard hyperbolic metric. Looking at the $\left(\Bbb{...
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1answer
52 views

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective?

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is a continuous function. I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(...
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1answer
92 views

$\phi:M\to \mathbb{R}$ continuous, $\phi(x)<\epsilon$ for $x\in X$, then $\phi(x)\le \epsilon$ for $x\in\overline{X}$

I was reading a proof that if a sequence of functions from $M$ to $N$, where $N$ is complete, converges uniformly in $X$, then they converge uniformly in $\overline{X}$, and it uses this result: $\...
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1answer
69 views

if $M$ is compact, then every continuous bijection $F:M\to N$ is an homeomorphism

My book proves that: if $M$ is compact, then every continuous bijection $f:M\to N$ is an homeomorphism by the following: Being $f$ closed, your inverse $g:N\to M$ is a function such that $F\subset ...
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0answers
32 views

Question about proof ot Tychonoff's theorem for metric spaces

Tychonoff's theorem: The cartesian product $M = \prod_{i=1}^{\infty}M_i$ is compact $\iff$ each $M_i$ is compact. My book, before proving it, says that the proof will happen like this: Given an ...