Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Equivalent definition of Cauchy sequence

A sequence $x_i$ is Cauchy if for all $r>0$, there exists $n$ s.t. $i,j\geq n$ implies $d(x_i,x_j)<r$. My question is, is it equivalent to define Cauchy as follows? $x_i$ is Cauchy if for all ...
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50 views

If $f$ is locally Lipschitz on $X$ and $X$ is compact, then $f$ is Lipschitz on $X$.

If $f$ is locally Lipschitz on $X$ and $X$ is compact, then $f$ is Lipschitz on $X$. My proof: Since $f$ is locally Lipschitz on $X$, for each $x ∈ X$ there exists an open $B_x$ containing $x$ such ...
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53 views

Existence of an Isometry of $\mathbb{R}^n$ onto Itself

For $a,b \in \mathbb{R}^n$, prove that there is an isometry of $\mathbb{R}^n$ onto itself which maps $a$ to $b$. I am working with the following definition of an isometry: Metric spaces $(X,d)$ and ...
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44 views

Detail in proof $(A \cup B)'=A'\cup B'$

Let $(X,d)$ be a metric space. Given $A\subset X%$, denote by $A'$ be the set of all limit points of $A$ in $X$. Now, I know several ways to prove that $(A\cup B)'=A'\cup B'$ which I'm perfectly ...
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42 views

Is $W$ connected or nowhere dense?

Let,$W\subset \mathbb R^{n}$ ba a linear subspace of dimension at most $n-1$. Which of the following statement(s) is/are true? (a) $W$ is nowhere dense. (b) $W$ is closed. (c) ...
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105 views

Decompositions of open sets in $\mathbb{R}^n$

This may be a basic question. It's well known that open sets in $\mathbb{R}$ are the union of disjoint open intervals. Does it similarly hold that open sets in $\mathbb{R}^n$ are the union of disjoint ...
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24 views

Show diameter inequality?

Please if someone can help, show that $$diam(A\cup B)\le diam(A)+diam(B)+d(A,B)$$ Where $diam(A)=sup\{d(x,y)|x,y\in A\}$ and $d(A,B)=inf\{d(a,b)|a\in A,b\in B\}$ Where $A,B\subseteq X$ and ...
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54 views

Help with contraction function in balls

A contraction is a function $f:X\to Y$ that satisfies:$$d(f(x),f(y))\leq qd(x,y);0<q<1$$ Now if we have a ball $B(a,r)$ and a contraction $f:X \to X$ Then the necessary condition for the ...
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25 views

Evenly dividing a discrete space

Suppose I have a 2D discrete space with coordinates 0 to 19 in the horizontal direction and 0 to 19 in the vertical direction. Coordinates could then range from [0, 0] to [19, 19]. I can evenly ...
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1answer
14 views

Definition of open relative, can't understand explanation

I can't figure out this paragraph: $E$ is open relative to $Y$ if to each $p\in E$ there is associated an $r\gt 0$ such that $q\in E$ whenever $d(p,q)\lt r$ and $q\in Y$ Does this look like ...
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35 views

Let $K_n$ is a nonempty compact subset of $X$ and that $K_{n+1} \subset K_n$ for each $n \in \Bbb N$. Show that $f(K) = \bigcap _n f(K_n)$.

Let $f: X \to Y$ be a continuous map between metric spaces. Assume that $K_n$ is a nonempty compact subset of $X$ and that $K_{n+1} \subset K_n$ for each $n \in \Bbb N$. Let $K := \bigcap _n K_n$. ...
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2answers
34 views

Continuous function and dense set

Let $f: X \rightarrow Y $ and $g:X \rightarrow Y$ continuous functions and $(X,d_X),(Y,d_Y)$ metric space. Let $E$ by a dense subset in $X$. Prove that $f(E)$ is a dense subset in $f(X)$. If ...
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31 views

Characterize pseudometrizable topologies.

Are there interesting characterizations of topologies that admit a pseudometric?
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154 views

Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).

Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective). If $f$ is not onto then there exist a $p \in X$ ...
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20 views

Suppose $S^m$ is a contraction on a complete metric space $(X,d)$. I want to show that this implies $S$ has a unique fix-point. [duplicate]

Let $(X,d)$ be a complete metric space and let $S: X \rightarrow X$ be a mapping. Suppose there exist $m \ge 1$ such that $\underbrace {S^m = S \circ S \circ \dots \circ S}_{\text {m times}}$ is a ...
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49 views

Is this proof sufficient to show that a concave function of a metric is also a metric?

Given a function $f: [0, \infty) \rightarrow [0, \infty)$ that's concave with $f(0) = 0$ and $f(x) > 0, \forall x \in [0, \infty)$, so $f(tx + (1-t)y) \geq tf(x) + (1-t)f(y)$, how do I show that if ...
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1answer
42 views

Exemplification in special function

this question have 3 sections that are together. $ a)$ Give an example of a map that is open but not closed, and an example of a map that is closed but not open. $b)$Determine whether the ...
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34 views

Satisfying the Definition of a Metric

Let $S$ be any set. Let $X$ be the collection of all finite subsets of $S$. Show that $d: X \times X \to [0,\infty), \quad (A,B) \mapsto |(A \setminus B) \cup (B \setminus A)|$ defines a metric on ...
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61 views

Metric space of infinite binary sequences

Let $\Omega = \{0,1\}^{\mathbb{N}}$ be the space of infinite binary sequences. Define a metric on $\Omega$ by setting $d(x,y) = 2^{-n(x,y)}$ where $n(x,y)$ is defined to be the maximum $n$ such that ...
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44 views

Metric Spaces: Finite Subsets

I have the following question regarding metric spaces. The problem is from Fred Croom's book "Principles of Topology". Show that a finite subset of a metric space has no limit points and is ...
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33 views

Proving whether a function on the set of real sequences is a metric

Let $S$ be the set of all real sequences , for $(x_n) , (y_n) \in S$ , let $d \Big((x_n) , (y_n) \Big) :=0 $ if $x_n=y_n , \forall n \in \mathbb Z^+$ otherwise $d \Big((x_n) , (y_n) \Big) :=\dfrac ...
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121 views

Concrete differences between $\mathbb{R}^n$ and general metric spaces

My question is inspired by the structure of Royden's Real Analysis, which introduces measure theory and Lebesgue integration for $\mathbb{R}$ in its Part I and then reconstructs large portions of ...
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1answer
46 views

Proving that a metric is an ultrametric

We have a metric space defined as the following: Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set ...
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53 views

Show that if a subset $E$ of a compact metric space $X$ is compact in $X$, then it is closed in $X$.

I am self-studying Royden's Real Analysis; Exercise 58 of Section 9.5, "Compact Metric Spaces", asks: Let $E$ be a subset of the compact metric space $X$. Show that the subspace $E$ is compact if ...
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1answer
41 views

Find a countable set with the given property.

Given any extended-valued $f$ on $(-\infty,+\infty)$, prove that there exists a countable set $D$ with the following property. For each $t\in\mathbb R$, there exist $t_n\in D$, $t_n\to t$ such that ...
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58 views

Lemma for $d(p,q)$ to be a metric for $\mathbb{R}^1$.

I have this exercise where I have to show whether or not $d(p,q)$ is a metric for $X = \mathbb R^1$, and I have come up with this lemma to help me with it. I need help proving it (if it is true). The ...
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52 views

a countable dense subset of Lipschitz functions

Suppose $(X,d)$ is a metric space and let $\mathcal{L}$ be the space of bounded Lipschitz functions on $X$. Let $D$ be a countable dense subset of $X$ and consider the set of functions ...
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1answer
45 views

Diameter of an open ball in a normed space

This is probably a silly question, but I'm reading some class notes that have the following proposition: In general it's true that $\operatorname{diam}( B(x,r) ) \leq 2r $ but in a normed space ...
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79 views

Boundary of the boundary of a closed set?

I'm trying to prove that for a closed set $S$, $S\subset X$ and $(X,d)$ a metric space, $\partial(\partial S)=\partial S$, where $\partial$ is the boundary of S defined as $$\partial S=\{\forall x\in ...
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Which metric spaces are isometric to $(\mathbb{R}^n, d_E)$?

Which metric spaces are isometric to $(\mathbb{R}^n, d_E)$? Two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$ are called isometric, when an isometry between them exists. An isometry is an isomorphism ...
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1answer
141 views

Proving an Ultrametric Space and Its Properties

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= ...
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63 views

Prove that if ($d(x,y)=0$ iff $x=y$) and if $d(x,z)\leq d(x,y)+d(z,y)$ then $d$ is a metric

Let S be a set and d a function from $S \times S$ into $\mathbb{R}$ such that $d(x,y)=0$ if and only if $x=y$ and $d(x,z) \leq d(x,y) + d(z,y)$ for all $x,y,z \in S$. Show that d is a metric. Here, ...
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16 views

Could anyone clarify the concept of translational invariance?

Tonight I stumbled across the concept of translational invariance while studying metric spaces, but I still do not have a clear conception of what it exactly means in abstract terms. Could anyone ...
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183 views

Is this map uniformly continuous? continuous?

Let $s$ denote the metric space of all sequences of complex numbers with the metric $$d(x,y) \colon= \sum_{j=1}^\infty \frac{1}{2^j} \frac{\vert \xi_j(x) - \xi_j(y)\vert}{1+\vert \xi_j(x) - \xi_j(y) ...
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1answer
26 views

The restriction $f|_A$ of $f$ to $A$ is a continuous function from the metric space $(A,d)$ to $(Y,d)$.

Let $f : (X,d) \to (Y,d)$ be continuous. let $A \subset X$ be open. Show that the restriction $f|_A$ of $f$ to $A$ is a continuous function from the metric space $(A,d)$ to $(Y,d)$. Please Help I am ...
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1answer
67 views

How to get an open ball in $[0,1]$ that contains $[0,1]$?

The definition of bounded we have is that if $X$ is a metric space, $z \in X$, and $X \subseteq X$, then there exists an open ball $B_z(R)$ with finite radius $R$ of $X$ centered at $z$ such that $X ...
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Relation between Lebesgue measure and nowhere denseness

As the title suggests, I am intereseted in knowing some relations between Lebesgue measures and the property of being nowhere dense, for subsets of R. An easy reult is that: Any closed set with 0 ...
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1answer
34 views

Is the sum of two complete metrics complete?

Let $X$ a space with two complete metrics $d_1$, $d_2$: Is $d=d_1+d_2$ complete?
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35 views

Difference between talking about collection $\{G_\alpha\}$ of open sets and finite collection of $G_1,\dots,G_n$ of open sets

Question: What is the difference between talking about "Any collection $\{G_\alpha\}$ of open sets" and "Any finite collection of $G_1,\dots,G_n$ of open sets"? I imagine they are highlighting ...
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1answer
36 views

If $f: X \to Y$ is a continuous surjection (onto), then a map $g: Y \to Z$ is open if $g \circ f : X \to Z$ is open.

If $f: X \to Y$ is a continuous surjection (onto), then a map $g: Y \to Z$ is open if $g \circ f : X \to Z$ is open. Since $g \circ f $ is open so image of every open set is open. Let $A$ be an open ...
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113 views

Every singleton set is open.

Prove that in the metric space $(\Bbb N ,d)$, where we define the metric as follows: let $m,n \in \Bbb N$ then, $$d(m,n) = \left|\frac{1}{m} - \frac{1}{n}\right|.$$ Then show that each singleton set ...
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3answers
58 views

Show every subspace of $\mathbb R^n$ is closed with respect to the usual metric?

How do I see that every subspace of $\mathbb R^n$ is closed with respect to the usual metric $p(x,y) = x^Ty$ ? I've seen some sweet results regarding Hilbert Spaces $\mathcal H$, especially that for ...
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1answer
66 views

Preserved properties through continuous linear maps

I just looked at the fact (at least according to Definition 2.8.1. in Distribution Theory by Friedlander et al.) that for $K_0\subseteq{\bf R}^{n(0)}$ compact, $\Omega_1\subseteq{\bf R}^{n(1)}$ open ...
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3answers
33 views

To show that the function $f(x) = \inf \{d(x,x_n) : n \in \Bbb N \}$ is uniformly continuous on X.

Let $(x_n)$ be a sequence in a metric space $(X,d)$. Show that the function $f(x) = \inf \{d(x,x_n) : n \in \Bbb N \}$ is uniformly continuous on X. I have started the proof in this way... Let ...
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4answers
57 views

Non-empty finite point set is closed

Subset of $\Bbb R^2$: My book says that non-empty finite point sets are closed. Why is this? Since it is a finite point set, it necessarily has no limit points within it, since every neighborhood of ...
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1answer
125 views

Is $C(\mathbb R)$ Separable?

I'm working on an exercise from Carother's chapter11 of Real Analysis that talking about Space of Continuous Functions: Here, $C(\Bbb R)$ is the set of continuous real-valued functions on $\Bbb R$, ...
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1answer
80 views

Is $C(\mathbb R)$ Complete?

I'm trying to prove an exercise from Carthers' book chapter10 of Real Analysis, problem claimed as, where $C(\mathbb R)$ denote the infinity norm space of all continuous functions on real line. I ...
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1answer
19 views

Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases? Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
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1answer
43 views

Is set $\{xy=1\}$ connected set in $4$-dim complex plane

Is a set $\{xy=1\}$ is connected in $4$-dimensional complex plane (over real number set).
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1answer
36 views

Metric under which $C([0,\infty );\mathbb{R})$ is a Polish space.

Does there exist a metric such that $C([0,\infty ); \mathbb{R})$ is a separable complete metric space? The usual supremum norm isn't even a metric on this space and I've tried several variants e.g. ...