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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Closed set, open set or neither?

Just a quick question - is a straight line that goes on indefinitely viewed as a closed set, open set or neither? Seeing as it includes all the boundary points as it travels, but it doesn't have any ...
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Quotient metric space

Let $X$ be some set, $(Y, \rho)$ be a metric space and $f:X\to Y$ be some map. Let $d$ be a pseudometric on $X$ defined by $d(x', x'') = \rho(f(x'), f(x''))$ and consider a quotient metric space $\...
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Completeness of “weighted” shortest path metric

I am trying to see when this type of metric is complete: Let $A$ be the set of $C^{1}$ paths in $U \in \mathbb{R}^{n}$. For any $x,y$ define $$\rho(x,y) = \inf_{\gamma \in A; \gamma(0) = x, \gamma(1) ...
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Proof for sets and functions.

I have been proving problems like this all day with ease, but this is is just puzzling to me. Where do I start? Also, a site with questions and answers to problems like these.
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Is an open ball a complete metric space?

Is an open ball $K((0,0),1) \subset \Bbb R^2$ with maximum metric a complete metric space? While I believe I understand basic metric space concepts I just don't have an idea how to prove or ...
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Is there a metric on the extended reals which yields regular and infinite limits?

The question is in the title: Is there a (extended) metric on the extended reals which yields regular and infinite limits? but in particular I want know the explicit construction of said ...
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Let $A$ be a subset of $\Bbb{R}$ such that the following $7$ sets are all different [closed]

I am suppose to come up with an example of an subset $A$ such that the sets $ A$ $int(A)$ $cl(A)$ $ cl(int(A))$ $ int(cl(A))$ $int(cl(int(A)))$ $ cl(int(cl(A)))$ are all different. I am ...
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Prove that cl(int(cl(int(A)))) = cl(int(A))

I am suppose to show that $ cl(int(cl(int(A)))) = cl(int(A)) $ and also that $int(cl(int(cl(A)))) = int(cl(A)) $ and I am having problems doing that becuse i just cannot figure out were to ...
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What is a contraction on a space $(X,d)$?

I have been reading some proofs on the elementary theorems of differential equations. One such proof uses the concept of a "contraction". See the definition below. Definition 4 Let $(X,d)$ be a ...
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show that $x \in A^o$ if and only if $d(A^c,x) > 0$

show that $x \in A^o \iff d(A^c,x) > 0$ where $d(A^c,x) = \inf_{y\in A^c} \lambda (x,y)$ where $\lambda$ is a metric and $(X,\lambda)$ is a metric space and $A^o$ is the set of interior points of A ...
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Connection between weak topology in probability and weak* topology in functional analysis

In functional analysis, Definition A: for any normed linear space $(X, \| \cdot \| )$, the weak star topology $\sigma (X^*, X)$ on $X^*$ is generated by the collection of seminorms $\{ p_x \,...
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written set of functions as a union of Borel measurable set

Denote by $\mathcal{H}$ the set of bounded and continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. I wonder if you can write $\mathcal{H}$ as (not trivial) $F_{\sigma}$ set in $...
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example showing Minkowski distance with $p<1$ is not a metric

The Minkowski distance: $$\left(\sum_i |x_i-x_i'|^p \right)^{1/p},\ \text{where}\ p\ge1$$ is only a metric for $p\ge1$. Can someone give me a quick example why the triangle inequality doesn't hold in ...
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Finding all metrics of set $X=\{1,2,3\}$

I have the following problem where I'm lost a bit. Let $X=\{1,2,3\}$ and $(X,d)$ be a metric space. List all the metrics $d$ of $X$ and show that they are equivalent. (Hint: construct a $3\...
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Proving that if a function is a metric then it is symmetric and non negative

I am trying to prove that given a metric d using only the properties that it $d(a,b)=0 iff a=b$ and $d(a,c)\le d(a,b)+d(b,c)$ that $d(a,b)=d(b,a)$ and $d(a,b) \gt 0$ I understand that it is part of ...
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Which metric to use to make the sequence 1, 1.4, 1.414, 1.4142, .. converges in space Q?

In space Q, with the metric it inherits from R, the sequence 1, 1.4, 1.414, 1.4142, ... does not converge. Is there a way to change the metric to make it converge in Q?
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show that $d(X,x) = d([X],x)$

show that $d(X,x) = d([X],x)$ where $d(X,x) = \inf_{y \in X} \lambda (x,y)$ where $\lambda$ is a metric and $[X] = \{ x \in X : d(X,x) = 0 \}$ I have shown $d([X],x) \leq d(X,x)$ I am stuck proving ...
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Is Topological Space a Metric Space?

What's the correct relationship between these two spaces? I think that topological space is a metric space, since the open is defined by a metric such that $d(x, a) < \epsilon$.
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Why is $[-1,1]$ compact when $a_n = (-1)^k$ does not converge in $A$

I know this question sounds silly but I was reading the definition of compactness and couldn't quite wrap my head around this Compactness :A subset $A$ of a metric space $M$ is compact if every ...
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Is $T$ a homeomorphism?

Let $X$ be the space of all polynomials in one variable over $\Bbb R$. If $p=a_0+a_1x +a_2 x^2+...+a_n x^n$,define $||p||=|a_0|+|a_1|+...+|a_n|$. Which are correct? $(X,d)$ is complete where $d(x,y)...
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Why is the function $f(x)=x^2$ is not a contraction on $[0,0.5]$?

Let $(X,d)$ be a metric space and let $F:A(\subset X)\to X$. We say $F$ is a contraction if there exists $\lambda$ where $0\leq\lambda<1$ such that $$d(F(x),F(y))\leq\lambda d(x,y)$$ for all $x,y\...
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How can one measure distance between point and the line in maximum metric space?

Given metric space $M = (\mathbb{R}^2, d)$ where $d = \operatorname{max}\{|x_1 - y_1|, |x_2 - y_2|\}$, how can one measure distance from some arbitrary point $X$ to the line $y = 3$, let's say? How ...
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If $X \setminus A$ is disconnected then prove or disprove $X \setminus B$ is also disconnected

Let $X$ be a connected metric space ( with more than one point ) and $A \subseteq X$ be not closed in $X$ and such that $X \setminus A$ is not connected ; then is it true that $X \setminus B$ is also ...
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Distance geometry and pythagorean theory. Pairwise distances to absolute 2D coordinates

I don't have sufficient mathematical background. I am trying to get the absolute 2D coordinates from the pairwise comparison distances: What I have distances between points: p1-p2 = 0.3 p1-p3 = 0.5.....
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Proving that $d(a,b)=p^{-n}$ is a metric for $\mathbb{Q}$

I have the following task: If we have the metric $d:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{R}$, so that $d(a,a)=0$ and $d(a,b)=p^{-n}$ always when $a-b=p^nh/k$, where $h,k,n\in\mathbb{...
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Why is the metric on $\mathbb{N}$ defined as the following?

This is from Muscat's Functional Analysis:http://staff.um.edu.mt/jmus1/metrics.pdf Show that $d(m,n) = |\dfrac{1}{m} - \dfrac{1}{n}|$, $m,n \in \mathbb{N}$ So the first two properties of the metric ...
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Does every connected metric space $X$ contains a connected subset $A$ such that $X \setminus A$ is infinite?

Convention : Whenever we are going to talk about connected spaces , we will mean with more than one point . I am trying to see whether every connected metric space $X$ contains a connected subset $A$...
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Existence of a special kind of continuous injective function $f\colon A \to \mathbb R$, where $A$ is countable, relating to connectedness

Let $A \subseteq \mathbb R$ be a countable set ($A$ induced with usual subspace topology), then does there necessarily exist a continuous injective function $f\colon A \to \mathbb R$ such that for ...
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How to prove $(0,1) \times \mathbb{R} \, , \, (0,2) \times \mathbb{R}$ are not isometric?

I am trying to prove in an elementary way that $X_1=(0,1) \times \mathbb{R} \, , \, X_2=(0,2) \times \mathbb{R}$ (with the standrad euclidean metric inherited from $\mathbb{R}^2$) are not isometric as ...
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Is the boundary of every compact connected subset ( with more than one point ) of $\mathbb R^2$ is Homeomorphic with $S^1$ ?

Is it true that the boundary of every compact connected subset ( with more than one point ) of $\mathbb R^2$ is Homeomorphic with $S^1$ ? I was thinking that union of two closed balls touching ...
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Isotropic Metric on $\mathbb{R}^{\mathbb{N}}$

I don't know if the term isotropic is correct in this context, but I was wondering if there exists a non trivial metric $\rho $ in $X=\mathbb{R}^{\mathbb{N}}$ such that $$\forall i \in \mathbb{N} \...
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Topologically-equivalent/metrically-equivalent metrics and the same topology

Definition: Metrics $d_1$ and $d_2$ on $X$ are topologically equivalent iff $d_1(x,x_n)\to 0 \iff d_2(x,x_n)\to 0$ for every $\{x_n\}\subset X$ and $x\in X$. Definition: Metrics $d_1$ and $d_2$ on $X$...
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about shape of open ball in metric space

consider following shape in plane : now we have definition of open ball in every metric space : $$B_r(x_0) := \{x \in X : d(x_0,x)<r\}$$ that radius is $r$ and center is $x_0$ , my questions are ...
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I want to prove that $f$ is continuous if its graph is closed

This is an exercise from Rudin's 'Functional Analysis': Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph $f:X\to K$ is a closed subset of $X\times K$. Prove that ...
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Open and connected set in metric space [duplicate]

In a normed space, we know that if a set is open and connected, it is path connected. Is it true for general metric space or general topological space?
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distance between a real and R\Q

Please how to prove that $d(x, R\setminus Q)=0, d(x,Q)=0$ for all $x\in \mathbb{R}$ ? I know that $d(x,R\setminus Q)=\inf_{a\in R\setminus Q} d(x, a)$ but how to continue ? Can i say that As $\...
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Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n (q_k-p_k)^2]^{1\over 2}$.

Let $p=(p_1,p_2,...,p_n)$ and $q=(q_1,q_2,...,q_n)$ be points in $\mathbb{R^n}$ with $p_k<q_k$ for each $k$. Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n ...
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Zero-dimensional separable metric spaces

I have to prove that every separable metric space, which is zero-dimensional is isomorphic to a closed subset of the Baire space. Maybe I can use the Baire category theorem, but I don't know how.
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How to show $dist(p,S) = 0$, then there exists a sequence in $S$ converging to $p$

Let $S \subset (M, d)$, where $(M,d)$ is a metric space Let $dist(p,S) \equiv \inf\{d(p,s) | s \in S, p \in M\}$ I wish to show that if $dist(p,S) = 0$, then there exists a $(p_n)$ in $S$ converging ...
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Type of convergence of a Cauchy sequence of functions on a complete metric space?

Let $\{f_n\}$ be a Cauchy sequence of functions defined on a complete metric space $E$. Then $f_n \to f$ on $E$. What is the type of this convergence? Is it pointwise?
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Let $f$ be a function $f:[0,1] \to [0,1] \times [0,1]$ now can we find $f$ with following conditions?

Let $f$ is a function $f :[0,1] \to [0,1] ×[0,1] $ now can we find $f$ with following conditions ?: 1- f be continues and one to one . 2- f be continues and onto . 3- f be continues and one to one ...
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Is a Normed Vector Space Necessary to Prove Path Connectedness?

Path connectedness seems to be defined in a topological space, but can the existence of a path be proven without using the functions of vector addition, scalar multiplication and norm ? For example, $...
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Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$

Let $(M,d)$ be a metric space and define: $d' : M$x$M \rightarrow R$ Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$ I know that $d'$ defines a metric on M, since d is a ...
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Does nonexpansive property in H-norm imply nonexpansive in 2-norm?

Suppose $\|f(x) - f(y)\|_H \le \|x - y\|_H$. In other words, $f$ is nonexpansive in the norm with respect to positive definite H: $\|z\|_H = z^T H z$. Can we then say something along these lines: $$\...
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Am I making some mistake in proving that $S$ is dense subset of $C[0,1]$?

Consider the space $X=C[0,1]$ with its usual 'sup-norm' topology.Let $$S= \{ f \in X : \int_{0}^{1} f(t) dt \neq 0\}$$ Show that $S$ is dense in $X$ We note that convergence with sup norm is ...
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31 views

Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n\times n$ matrices with real entries

Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n \times n$ matrices with real entries. Definition 1.26. Let $X$ be a linear space (over $R$). A ...
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Retracts and dense $G_{\delta}$ sets

Suppose $X$ is a complete separable metric space and $Y\subset X$ is a retract of $X$ with retraction $r\colon X\longrightarrow Y$. I am interested in the following question: Given a dense $G_{\delta}$...
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Is the metric on the circle, induced from the plane, not a flat one?

My question concerns the highlighted part posted below, from Wikipedia article. (Link to the revision at the time of this post.) I'd say I can't detect the curvature of the unit circle if I go along ...
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Get a bounded metric from a metric - triangle inequality for $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ [duplicate]

This is related to Proof that every metric space is homeomorphic to a bounded metric space but I can remember that if $d$ is a metric, then $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ is also a metric that ...
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Non-separability of normed spaces

I would like some hints to decide when a normed space is separable or not. I really understood the definition and the classic examples of separable spaces but when I go to show that a space is non-...