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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Hausdorff dimension of Sierpinski triangle

https://en.wikipedia.org/wiki/Hausdorff_dimension#Behaviour_under_unions_and_products Wikipedia page says that if $ \underset{i \in I}{\cup} X_i = X$ and $I$ is countable then $dim_{Haus}(X) = ...
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Does computing distance in $N$ dimensions have any application beyond $N=3$?

So I had to implement the distance formula earlier in programming, and had a thought that I should make it work for $N$ dimensions. I then smacked myself and realized that I'm programming for a ...
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$X$ is a normed linear space such that for some compact $K\subseteq X$ , $\operatorname{span} K$ is dense in $X$ then is $X$ separable?

Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and ...
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If $X$ is a compact metric space and $E_n$ is closed nonempty subset, show that $\cap_{n=1}^\infty E_n$ is nonempty.

Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is ...
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How limiting/ heavy is the “triangle inequality” assumption?

Suppose a theorem proves something about a family of distance measures, with this the triangle inequality assumption. How limiting this assumption is in reality? What are some real-world examples of ...
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36 views

Proving $\mathbb{R}/\sim$ is homeomorphic to unit circle

Let $S$ be the unit circle in $\mathbb{C}$, standard topology. Define the equiv. rel. $\sim$ on $\mathbb{R}$ as $x\sim y\iff x - y\in\mathbb{Z}$. I would like to prove that $\mathbb{R}/\sim$ is ...
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40 views

Continuity of a function between metric spaces

I want to show: Let $(X,d)$ be a metric space and $A \subset X$ be a closed subset. Define $f: X \to \mathbb{R}$ by $$ f(x) = d(x,A) := \inf_{y\in A}d(x,y), \phantom{.} \forall x \in X.$$ Show ...
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Euclidean metric in $\mathbb{R}^n$; the singleton is not open in such a metric space

I am trying to prove this but just don't see it. We are talking about openness in the metric sense, yes? So, my attempt is Let $x \in \mathbb{R}^n$ and $d$ represent the Euclidean metric, ...
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Is $\{\frac{m}{10^n}\mid m,n\in\mathbb Z,\quad n\geq 0\}$ dense in $\mathbb R$?

The set $S$ of real numbers of the form $m/(10^n)$, $m,n$ integers and $n$ greater than equal to $0$, is dense sunset of $\mathbb R$ or not?? I know dense means closure of $S$ in $\mathbb R$ is ...
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An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and ...
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72 views

Property of Nowhere Dense Sets

I am trying to prove the following statement regarding nowhere dense sets: "In a metric space X, the frontier of an open set is the set of accumulation points of a discrete set." As far as my ...
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What simple topological properties of conic sections can be explored?

In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them? Edit: To clarify, ...
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$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
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Topological spaces that remain non-metrizable, if the definition of metric space allows $d(x,y) = 0$ where not necessarily $x = y$?

In the definition of metric space, only one thing strikes me as unnatural: the requirement that $d(x,y) = 0$ implies $x = y$. As a programmer, I don't find it uncommon to deal with equivalence ...
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Give a “constructive” proof of the fact that in a metric space the intersection of two open balls is open

Main Question Can someone give a "constructive" proof of the fact that, Let $(X,d)$ be a metric space and $x,y\in X$. Let $B_d(x,r_x)$ and $B_d(y,r_y)$ be two open balls centered respectively at ...
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Proof that a subset of metric space with euclidian norm is open iff the same subset is open in metric space with Manhattan norm

For $\mathbb{R}^2$ we have the euclidian norm $$(x_1,x_2)\mapsto\sqrt{x_1^2+x_2^2},$$ and the Manhattan norm $$(x_1,x_2)\mapsto|x_1|+|x_2|.$$ Let $d_E$ and $d_M$ be the metrics defined by these norms, ...
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49 views

Open or closed set in $\mathbb{R}$

I have this set $A=\left\{\frac{1}{n}|n\in\mathbb{N}\right\}$ I need to show that it is neither open or closed in $\mathbb{R}$. And that the union ...
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Functions between metric spaces (and how they relate to closures of sets)

Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$ we have $f(\overline{A})\subset \overline{ f(A) }$ ...
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37 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
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+100

Preservation of completeness through a continous onto mapping

Let $(X_{1},d_{1})$ and $(X_{2},d_{2})$ be metric spaces and $f: X_{1} \to X_{2}$ be a continuous onto map such that $$ d_{1}(x,y) \leq d_{2}(f(x),f(y)) \hspace{2mm} \forall\phantom{i}x,y \in ...
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Interior of a set in a metric space

if $E$ is a metric space nd $B\neq E$ how to prove that: $$\overset{\circ}{B}=\bigcup_{n=1}^{\infty} (\{x\in E, d(x, E\setminus B)\geq \frac1n\})$$ i don't know how to start
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18 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
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Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
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22 views

For $U\subseteq Y\subseteq X$, prove that $U$ is open in $Y$ iff there is a $V\subseteq X$ such that $U=Y\cap V$

Let $(X,d)$ be a metric space, with $Y$ a subset of $X$. How do I prove that a subset $U\subseteq Y$ is open in the metric space $(Y,d|_{Y\times Y})$ iff there exists an open subset $V$ of $X$ ...
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40 views

If $\sum_{n=1}^{\infty}x_n^2<\infty$ and $\sum_{m=1}^{\infty}x_n^2<\infty$, is $\sum_{k=1}^{\infty}(x_n)_k^2(x_m)_k^2<\infty$? [duplicate]

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ Prove that $l^2$ is complete with ...
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Prove that all three metrics induces the same topology on $X_1\times X_2$

Prove that if $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces on $X_1\times X_2$ and metric $d:(X_1\times X_2)\times (X_1\times X_2)\rightarrow R$ is defined in following way: ...
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How to resolve the apparent paradox resulting from two different proofs?

Definition of Open Ball Let $(X, d)$ be a metric space and let $r\in\mathbb{R}^+$. Then the set, $B_d(x, r) := \{y \in X : d(x, y) < r\}$ will be said to be the open ball of radius $r$ ...
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141 views

Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in ...
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1answer
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Function between two metric spaces?

I need to come up with: two metric spaces ( X , d ) and ( Y , p ) A continuous function f: X → Y A Cauchy sequence {xn} in X that isn't mapped to a Cauchy sequence in Y My idea was to make ...
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Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
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What values of $b$ such that $f_n(x)=b\cos\left(\frac{x}{n}\right)$ converges uniformly?

For what values of $b$ does the sequence of functions: for each $n\in\mathbb{N}$, let $$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$ converge uniformly in the space $C[0,1]$ equipped ...
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+100

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
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A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
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Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...
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subset of a complete space has a relatively compact $\varepsilon$-net.

Trying to prove that, a subset $A \subset X$ of a complete space $X$ is relatively compact iff $\forall \epsilon > 0$ $A$ has a relatively compact $\epsilon$- net. I have proved the following ...
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Definition of topological space

The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the ...
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Can we define the derivative of a function in arbitrary metric space in the following way?

Let us first define some terms. Definition of Pre-pseudometric Let $X\ne\emptyset$ and a function $\varphi:X\times X\to\mathbb{R}$ will be called a pre-pseudometric on $X$ if, ...
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Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
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How can I show that the sequence $a_n := p^n$ is a convergent sequence in this metric and find its limit?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
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Equicontinuous homotopies of families of uniformly equicontinuous functions

Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of ...
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Is there a term for this line like subset within a metric space

While thinking about geodesic lines I started exploring subsets of a metric space that have the following property. $ \forall a,b,c \in L, d(a,c)> d(a,b) \land d(a,c) > d(b,c) \implies d(a,c) = ...
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If $Z(f)$ is the zero set, prove that $Z(f)$ is closed

Introduction: Exercise from Principles of Mathematical Analysis, third edition (Rudin), page 98. Exercise: Let $f$ be a continous real function on a metric space $X$. Let $Z(f)$ (the zero set of ...
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72 views

Is the empty set an open ball in a metric space?

Problem Let $(X,d)$ be a metric space where $X$ is a non-empty set. Is the empty set an open ball in $X$? I think that it is true because if $X=\mathbb{R}$ with the usual metric then for all ...
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proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$

Intuitively this proposition seems true, but I've been told that is not a trivial thing to prove. Is there any simple proof (or counter-example) for the proposition: Consider the closed ball of ...
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1answer
36 views

EDITTED: Find all values of $a$ and $b$ so that $ax^n+b\cos\left(\frac{x}{n}\right)$ is Cauchy.

For each $n\in\mathbb{N}$ let $$f_n(x)=ax^n+b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1].$$ Find all values of $a$ and $b$ for which $(f_n)$ is a Cauchy sequence in $C[0,1]$, the space of ...
3
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1answer
18 views

Any bi-invariant distance on a group is inverse-invariant?

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...
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51 views

Is every compact metric space finite dimensional?

It certainly is true if the metric space is normed by Riesz lemma.
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27 views

Equivalent Metrics on $\mathbb{R^n}$

I am working on a problem and want to verify that my logic and reasoning is correct. This is my first time working with metric spaces. Show that the following define equivalent metrics on ...
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34 views

Does a mapping from one metric space to another metric space preserve star-likeness of regions?

Let $X$ be a vector space, let $M_1 = \left(X, d_1\right)$ be a metric space and let $M_2 = \left(X, d_2\right)$ be another one. $f : M_1 \to M_2$ is continuous and the origin is a fixed point. $f$ ...
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1answer
15 views

Is a metric space a requirements for the application of the algebra of events from probability?

When I refer to a metric space, I mean a space that has some genuine notion of distance. In some applied context, this distance would be computed with respect to a coordinate system. I just wanted to ...