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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Metric space $(X,d)$ with distance $D(x,S)=\inf\{d(x,y)|y\in S\}$ for $S$ subset of $X$

Let $(X,d)$ be a metric space with $S$ a non-empty subset of $X$. For $x\in X$ we define the distance $D$ between $x$ and $S$ as $D(x,S)=\inf\{d(x,y)|y\in S\}$. How do I prove that $\overline{S}$ ...
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continuous map of metric spaces and compactness

Let $f:X\rightarrow Y$ be a continuous map of metric spaces. Show that if $A\subseteq X$ is compact, then $f(A)\subseteq Y$ is compact. I am using this theorem: If $A\subseteq X$ is sequentially ...
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Which subsets of $l^2$ are compact?

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ State whether the following subsets ...
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About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
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Proving that union of epsilon nets is an epsilon net

While reading a paper, I came across these definitions and claims: Definition: Given $p \in \mathbb R^d$, and $H$, a set of hyperplanes, let $$\text{Violate}_p(H) = \{h \in H: h \text{ is strictly ...
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$X$ sequentally Compact implies that $X$ is complete

I was reading through Roydens book and there is one part that I don't understand. Here is the proof. Suppose $X$ is sequentially compact metric space, then $$X \text{ is sequentally compact}:= ...
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Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
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A specific embedding of semisphere on $R^2$.

I was playing with piece of paper which has the form of semisphere, to be more precise we may assume that it satisfies $x^2+y^2+z^2=1$ for nonnegative $z$. I tried to make it flat without stretching ...
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are two metrics with same compact sets topologically equivalent?

are two metrics with same compact sets topologically equivalent ? I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set is ...
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Show that $|d(m,n) -d(n,o) | \leq d(m,o)$ for a metric space

Problem Let $(M,d)$ be a metric space. Show that $$|d(m,n) - d(n,o)| \leq d(m,o) \ \forall m,n,o \in M$$ Since $(M,d)$ is a metric space I know it fufills the triangle inequality. So if I ...
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What is the meaning of “infinitesimal structure”?

Reading a Differential Geometry book I found this sentence: "A main theme in analysis on metric spaces is understanding the infinitesimal structure of a metric space." I cannot understand the meaning ...
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In a complement of a closed set find a point that is closest to any point in the closed set

Is it true that for $U$ closed and nonempty in a metric space $X$, let $a\in X\setminus U$, then there exists a $b\in U$ such that $d(a,b)\leq d(a,x) \forall x\in U$? I think it is correct because it ...
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Is $\mathbb{R}$ an open ball in $\mathbb{R}$?

If we write $B(0,\infty)$ as the open ball then $\mathbb{R}$ is an open ball in $\mathbb{R}$. Is it correct?
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Is there anything wrong in the following proof?

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces and let $U\subseteq X$ and $V\subseteq Y$ such that $U$ and $V$ are respectively open in $X$ and $Y$. Show that $U\times V$ is open in ...
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Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...
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Connectedness proof of a metric space. [on hold]

Assume that there is a $p$ in a metric space $(\chi, d)$ such that the function $f(q) = d(p, q)$, $q \in \chi$ omits the value $c > 0$, but takes values greater than $c$. Show that $(\chi, d)$ is ...
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How many metrics are there on a set up to topological equivalence?

I want to find the number of topologically nonequivalent metrics on a set. I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set ...
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An example of the set of distances of two points in two different closed sets having no infimum

On a problem set for my Analysis in Several Dimensions class (basically real analysis on multivariable functions), I encountered this question: Let $(X, d)$ be a metric space, let $C ⊂ X$ be a ...
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Is every compact metric space hereditarily separable?

Let $X$ be a compact metric space. I see why all open and closed subsets of $X$ are separable. But is every subset of $X$ necessarily separable? EDIT: Since $X$ is separable metric, it embeds into ...
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How to define and compute the norm of a vector with riemannian metric?

Let us consider for example, the riemannian metric $g=e^xdx^2+dy^2$ (it is symmetric and definite positive), with associated matrix $\begin{pmatrix} e^x & 0\\ 0 & 1 \end{pmatrix}$. Consider ...
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Cauchy continuous implies standard continuity

Let $f$ be Cauchy continuous. $f$ is Cauchy continuous if for any Cauchy sequence $\{x_{n}\}$ in $(X,d_{X})$, $\{f(x_{n})\}$ is a Cauchy sequence in $(Y,d_{Y})$. Show that Cauchy continuous $\implies$ ...
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Metric spaces inside of metric spaces

Let $(X, d)$ be a metric space, $Y$ ⊂ $X$ and consider the metric space $(Y, d)$. Show that every open set $U$ in $Y$ has the form $U$ = $V$ ∩ $Y$ for an open set $V$ ⊂ $X$. Show that ...
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What is the max size of subset of a Hamming Space with this property

Keep in mind I know virtually no coding theory, I simply recognized this as an equivalent formulation to another question I was considering. Let $F(r,n)$ denote the set of all words of length $r$ on ...
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A metric between functions on $\mathbb{R}^2$

I want to measure the distance between functions $f$ and $g$ (not necessarily continuous) on a bounded subset $M\subset\mathbb{R}^2$. I assume $f$ and $g$ are locally integrable and bounded on $M$. ...
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$A,B$ closed subsets of $\mathbb R^n$ , when can we say (other than compact-ness of $A$ or $B$ ) $\exists b \in B$ such that $dist(A,B)=dist(b,A)$ ?

Let $A,B$ be disjoint closed subsets of $\mathbb R^n$ , when can we say ( weaker than compact-ness of $A$ or $B$ ) that there exist $b \in B$ such that $dist(A,B)=dist(b,A)$ ? I know that if $A,B$ are ...
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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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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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I know the basic definition of continuity. But here, the definition is applied for a ball.

I am studying the topology of $\Bbb R^n$ from W. R . Wade's Introduction to analysis book. I know the basic definition of continuity. But here, the definition is applied for a "ball". I dont ...
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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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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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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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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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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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Why is $U ⊂ \mathbb{R}^n$ open with respect to metric $d_p$ iff it is open with respect to metric $d_q$ for $q ∈ [1, ∞)$?

Let's say that for any $p ∈ [1, ∞)$ we have a distance function on $\mathbb{R}^n$ given by $$d_p(x, y) := \left(\sum^n_{j=1}|x_i - y_i|^p\right)^{\frac{1}{p}}$$ How would I show that a set $U ⊂ ...
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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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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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Prove that the following statements are equivalent characterizations of continuity

Let $f: (X,d) \rightarrow (Y, d')$ be a function. Prove that the following are equivalent: $f$ is continuous . For every $A \subset X$, $f(cl(A)) \subset cl(f(A))$. For every closed set $B$ in ...
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Show that in a discrete metric space, every subset is both open and closed.

I need to prove that in a discrete metric space, every subset is both open and closed. Now, I find it difficult to imagine what this space looks like. I think it consists of all sequences containing ...
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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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Examples of decreasing sequences of closed sets with constant diameter and empty intersection in complete metric spaces

Looking through older exams from the topology class I'm taking, I found an interesting problem. Give an example: $ (X, d) $ - a complete metric space $ F_1 \subset F_2 \subset F_3 \subset ... $ - a ...
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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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If two nested open sets have the same nonempty boundary, are they the same set?

Let $(X,d)$ be a metric space. Let $B_\epsilon(x)$ be the open ball of radius $\epsilon$ centered at $x$. For $x\in X$ and $\epsilon>0$, suppose that $V$ is an open set in $X$ with $V\subseteq ...
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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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In a standard metric space…what does | | mean (is it the absolute value or something more)?

We have a standard metric space defined as: ($\mathbb{R}$,d)= ($\mathbb{R}$, | |) $d(x,y)=|x-y|$ Does | | in first sentence always mean that we must do $|x-y|$; so that we look only at the distance ...
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If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$, does it follow that $A$ is dense?

If $d(x,A)=0 \:\:\forall x\in X$ for some subset $A$ of $X$ then $A$ is dense in $X$, right? Once I did one problem which says $d(x,A)=0\Leftrightarrow x\in \bar{A}$ so by the condition here we get ...
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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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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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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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How to show that continuous functions between metric spaces agree on a closed set

Let $(X,d)$ and $(Y,d')$ be metric spaces, and let $D$ be a dense subset of $X$. Show that: If $f:X\to Y$ and $g:X\to Y$ be continuous, then the set $\{x\in X\mid f(x)=g(x)\}$ is closed.