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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A map between metric spaces preserving convergent sequences is continuous

Pugh, "Mathematical Analysis", exercise 2.17: Assume $f : M \to N$ is a map from one metric space to another which satisfies the following condition: for every convergent sequence $(a_n) \subset ...
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Show that $\rho : X \times X \mapsto \mathbb{R} $ is continuous function on $(X \times X, \tau)$

Show that $\rho : X \times X \mapsto \mathbb{R} $ is continuos function on $(X \times X, \tau)$ where $\tau ((x_1,x_2), (y_1,y_2)) = \sqrt{\rho (x_1-y_1)^2 + \rho (x_2-y_2)^2}$ and $X \times X$ is the ...
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$X\subset Y \implies \overline{X}\subset \overline{Y}$ (closure inclusion subset)

Suppose that $X,Y\subset M$, being $M$ a metric space. In order to prove that: $$X\subset Y \implies \overline{X}\subset \overline{Y}$$ If $x\in \overline{X}$ we have that $d(x,X) = 0$. But I ...
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prove that $(X\cap Y)^- \subset X^- \cap Y^-$

I have to prove: $(X\cap Y)^- \subset X^- \cap Y^-$ Well, if $a\in (X\cap Y)^-$ then there is an open set $A$ containing $a$ such that: $$A\cap (X\cap Y)\neq \emptyset$$ I've tought of some ...
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Proving $(A\times B)^- = A^-\times B^-$ (closure of cartesian product)

My proof, for: $$(A\times B)^- = A^-\times B^-$$ using the metric $$d''((a_1,a_2),(b_1,b_2)) = max\{d_1(a_1,b_1),d_2(a_2,b_2)\}$$ $\rightarrow$ Well, if $a = (a_1,a_2)\in (A\times B)^-$ then: ...
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Understanding distance between point and line via infimum

The distance between point and line independently on metric is defined by $$d(X, l) = \inf\{d(X, Y)|Y\in l\}.$$ I have troubles understandning how this infimum works. Can someone please give me an ...
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Proof of $\partial\partial\partial S=\partial\partial S$

How can I prove in metric space that $$\partial\partial\partial S=\partial\partial S$$ using the proposition of the boundary below? $$\partial S=\text{cl}S \cap \text{cl}(S^c)$$ I found that if ...
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Are these metrics?

I want to find if the below functions are metrics. I have worked through each of the three conditions, but am stuck on the positivity of $f(a, b)$ (first condition-see below) and the triangle ...
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If a set is separable its completion is separable

$\tilde{M}$ is the completion of M. This question has ben done before, but with no answer. My idea is to first: show that $S$,$T$ sets (which compose the separation of M) are actually the preimage of ...
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Are $(C[0,1],d_\infty)$ and $(C[0,1],d_1)$ homeomorphic?

Two metric spaces are said to be homeomorphic if there is a bijection f between them such that $f$ and $f^{-1}$ are both continuous. Consider $C[0,1]$ with metrics: $d_\infty (f,g)=\max_{x\in ...
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Showing that $X$ is locally compact

Now I would like to change question in If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. Is it true that If $X$ is Hausdorff, second ...
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Can every compact subset of $\Bbb R^n$ be written as a disjoint union of compact subsets, where each of them are path-connected?

I was wondering if every compact subset of $\Bbb R^n$ could be written as a disjoint union of compact subsets, where each of them are path-connected, i.e. : If $X \subset \Bbb R^n$, $n \ge 1$, $X$ is ...
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Cannot prove that open balls are not open sets in $\kappa $ metric space .

$(X,d)$ is a $\kappa $ metric space . $x\in X$. Take the open ball $B_d(x,r)$ with radius $r.$ Consider $y\in B_d(x,r)$ s.t $d(x,y)={{4r}\over 5}$ In the usual case , the ball $B_d(y,{d\over 5})$ is ...
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The $Hol$ operator is a continuous function?

Let $\Omega$ be a compact space, and consider $C(\Omega)$ the space of the continuous functions over $\Omega$, consider also, $C^\gamma(\Omega)$ the space of all $\gamma$-holder continuous ...
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$\overline M = M$ (closure of $M$ is $M$)

I have an intuition that an entire metric space (not a subset of it) does not have boundaries, so its closure would be itself, but how to prove it?
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Show that U is open set in metric space $(R^2,d_1)$ if and only if U is open set in metric $(R^2,d_{\infty})$

Show that U is open set in metric space $(R^2,d_1)$ if and only if U is open set in metric $(R^2,d_{\infty})$ $d_p(x,y)=(\sum^n_1 |x_i-y_i|^p)^\frac{1}{p}$ in $R^n\\$ $d_{\infty}(x,y)=max^n_{i=1} ...
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Locally compact Hausdorff space is metrizable

Given $X$ a Hausdorff space, I have a hunch that $X$ is locally compact $\iff X$ is metrizable. I am not sure if it is true because I do not know how to prove that. To prove the implication ...
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Are these two metric spaces complete?

Let $J$ be an arbitrary non-empty set of indices, and let $\mathbb{R}^J$ denote the set of all the $J$-tuples of real numbers (i.e. the set of all the functions $x \colon J \to \mathbb{R}$, where we ...
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Proving “I-divergence implies I* - divergence” when (AP) condition is satisfied in $\kappa$-metric space

The Definitions : I-divergence :- A sequence $\{x_n\}_{n\in \mathbb N}$ is called I-divergent if $\exists$ an element $x\in X$ such that for any positive real number $G$ , $A(x,G)=\{n\in ...
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$d(a,X) = 0 \iff X\cap U\neq \emptyset$ (distance from set equals 0 iff is adherent point)

I need to prove that: $$d(a,X) = 0 \iff X\cap U \neq \emptyset$$ for all open set $U$ that contains $a$ My idea is that if $d(a,X) = 0$, then there is a point $b\in X$ such that $d(a,b)=0$. In some ...
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A non metric first countable topological space [duplicate]

Every metric space is first countable, but what about the converse? Does it always hold? If not, can anyone give a counterexample? Thanks
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Is this a valid definition of “self-similar fractal”?

I have always been fascinated by self-similarity, particularly in fractals. I was always wanted to find a simple definition of a self-similar fractal. Of course, saying "is self-similar, and is a ...
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Connectedness of $A\cup B$ [duplicate]

Show that the following set is connected subset of $\mathbb R^2$ : $$\left\{\left(x,\sin \frac{1}{x}\right)\in \mathbb R^2|0<x<\infty\right\}\cup\left\{(0,0)\right\}$$ Attempt : Here ...
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Non empty set with zero diameter

Let $A \subset X$ where $X$ is a metric space. by definition diam$(A) = \sup\{ d(x,y), x,y \in A\}$. if $A$ is non empty and has zero diameter, can I conclude that $A$ is a singleton? i reason as ...
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Proving the Hausdorff property for $\kappa$-metric spaces

I want to prove that the Hausdorff property holds for all $\kappa$-metric spaces. For $\kappa \neq 1$, $(X,d)$ is a $\kappa$-metric space if $X$ is a set and $d$ is a function $X\times X ...
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Showing $(\Bbb R^n,d)$ is complete.

Let $X$ be the space of all $n-$tuples $x=(x_1,x_2,\ldots,x_n)$ of real numbers. Define $$d(x,y)=\max_i |x_i-y_i|, \qquad \text{where } y=(y_1,y_2,\ldots,y_n).$$ Show that $(X,d)$ is complete. ...
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Compact subset continuity

Can someone help me with this? it seems really easy question but i couldn't see it through... For a compact Subset $K$ of a metric space $X$ and $x \in X$. The Function is $f:X\to R$ given by $f(x) = ...
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How can a continuous function induce a proper inclusion $f(\overline{A})\subsetneq \overline{f(A)}$?

Let $f:(X, d_X)\longrightarrow (Y, d_Y)$ be a continuous function between two metric spaces, $A\subseteq X$. We have $f(\overline{A})\subseteq \overline{f(A)}$ from this question. Can you please ...
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Triangle inequality in $\kappa $ metric space where $\kappa = 2^n $

$X$ is a $\kappa $ - metric space if $$d: X\times X \rightarrow \mathbb R$$ satisfies the following : $$a)\ \ d(x,y)\ge 0;\\b)\ \ d(x,y)=d(y,x);\\c)\ \ d(x,y)=0\ \ \iff\ x=y;\\d)\ \ d(x,z)\le ...
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Topological proof of the compactness of product metric space

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric spaces (see the definition here). Then show that the product metric space $(X\times Y,d_{X\times Y})$ is also compact. Now this can be ...
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Can the isometry group of a metric space determine the metric?

Let $(X,d)$ be a metric space. There are always other metrics on $X$ which generates the same topology, and have the same isometry groups, for instance $\tilde d =\sqrt d$. (The same will be true for ...
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paralell translations $P_{c}$ on manifold $M$ are $C^{\infty}$ diffeomorphism

According to the following image . Where . Question: How can i conclude from the above lemma that $P_c$ is a $C^{\infty}$ diffeomorphism? Thanks for your time.
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Given a function $f: X \to Y$, if $X$ is compact, prove the graph $g = (x, f(x))$ is compact in $X\times Y$

Given a function $f: X \to Y$, and graph of $f, g = \{(x, f(x)): x\in X\}$ in metric space $X\times Y$ (a) Suppose that $X$ is compact. Prove that $f$ is continuous if and only if $g$ is a compact ...
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Show that $f(U) = \big\{ f(x_1,x_2) : (x_1,x_2) \in U \big\}$ is an open set of $\mathbb{R}$ with the standard metric.

Let $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x_1,x_2)=x_2$. I am required to show that $$f(U)= \big\{ f(x_1,x_2) : (x_1,x_2) \in U \big\}$$ is an open set of $\mathbb{R}$ with the standard ...
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Showing two metrics are equivalent.

Let $(X,d)$ be a metric space. Define $$d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ (do you know the name of this metric?) Show that the metrics $d$ and $d_1$ are equivalent. Edited: Captain Lama pointed out ...
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How are neighbouring sequences defined? (Metric spaces)

What does it mean for sequences to be neighbours in a metric space? My attempt is: In a metric space $(X,d)$, $(x_n)$ and $(y_n)$ are neighbouring sequences iff $$\forall_{\epsilon>0}\exists_N ...
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Why if $p (x_1, \dots, x_n)$ is polynomial on $\mathbb{R}^n$, then $p (x) \neq 0$ is satisfied by open dense set?

I have problems in seeing what exactly is the all point of first category and second category sets. Finally, I've found a reference (Bredon's "Topology and Geometry") that introduces the topic in a ...
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Entropy of factor map

Let $A$ and $B$ be two compact metric spaces with $B\subset A$- Moreover, let $T\colon A\to A$ continuous and let $S\colon A\to B$ a continuous surjection with $S\circ T=T\circ S$. Moreover assume ...
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Show$d : x, y \in E \to \|x − y\|_E$ is a distance on $E$ and every open ball is the image of the unit ball under an affine map then check closure.

A norm $\| · \|$ $E$ on a $\Bbb R$-vector space E is a function from E to [0, ∞) such that: $\forall x \in E$ we have $\|x\|_E =0 \iff x=0$, $\forall x\in E$and$ \ λ \in \Bbb R$ we have $\|λx\|_E ...
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Estimate transition rate from vectorized state A to state B

Assuming that the conditions of a state are described by a vector where its coordinates represent the values of specific variables, could the distance between two states depict the transition rate ...
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Prob. 18, Chap. 2 in Baby Rudin: Any non-empty perfect set of real numbers which contains no rationals? [duplicate]

Here's Prob. 18 in the Exercises after Chapter 2 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Is there a non-empty perfect set in $\mathbb{R}^1$ which contains no ...
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Explicit homemorphism between code space

Given a finite set we can construct the string (or code)space $S^\omega$ with the metric $\rho(w,w')=r^m$ where $w,w'\in S^\omega$ and $m$ is the maximum number such that the two strings $w,w'$ ...
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Replacing “infinite” with “finite” of a statement

Statement: The intersection of a finite collection of open subsets of $R^n$ is open in $R^n$. Proof of the statement: Suppose that $\mathcal{O}=\bigcap_{i=1}^{k}\mathcal{O}_i$ for some $k$, ...
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How can i prove those metrics are the equivalent?

We got 2 metric spaces in $\mathbb{R}$, and these metrics: $d_1(x,y):= |x-y|$ and $d_2(x,y):= |x^3-y^3|$. I'm asked to prove this by proving that the identity which goes from one metric space to ...
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In any metric space X, the sets ∅ and X are always closed. [duplicate]

In any metric space X, the sets ∅ and X are always closed. Hello, I need help in proving this statement. I'm not exactly sure where to start. I have tried to teach myself the definitions, however it ...
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Isometries between the hyperboloid and the plane?

What is the map from $\mathbb{H}^2$ to $\mathbb{R}^2$ that preserves the pairwise geodesic distances in one as closely as possible to the pairwise geodesic distances of the images in the other? ...
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Prove the empty set is closed for a metric space (X,d).

I'm not sure how to approach the proof for this. I know this statement is vacuously true, because the empty set does not contain any accumulation points. But I'm having a hard time writing a proof for ...
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I've got a definition, but says something strange, what does it mean?

$M_1:=(M,d_1), \ \ M_2:=(M,d_2)$. $d_1$ is equivalent to $d_2$ if the identity $x\rightarrow x$ of $M_1$ over $M_2$ is an homeomorphism I'm not sure what it is talkin about when it says "identity" ...
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Help understanding a proof that the metric space of bounded functions is complete.

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...
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1answer
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Why does the second statement follow from the first one?

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...