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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Metric spaces - continuity - open/closed.

Let $f:(M_1,d_1)\to (M_2,d_2)$ be a mapping between two metric spaces. a)Let $A\subseteq M_1$ be open and $B\subseteq M_1$ closed. Show through the use of counterexamples that in general ...
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countable dense set of space of continuous functions on a campact set

Let $X$ be a compact metric space. Let $C_+(X)$ be the set of all non negative continuous functions on $X$. Do there exist a countable dense set of $C_+(X)$? I think the answer is affirmative. For ...
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42 views

Metric spaces as Cauchy complete categories, nlab entry, insight into a few of the constructions.

I'm having a bit of trouble making sense of some of the concepts in the "Metric space" section on nlab's entry on "Cauchy complete category" ...
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36 views

Existence of an homeomorphism between $X$ a complete separable metric space and a subspace of $[0,1]^{\mathbb{N}}$

Result: If $X$ is a complete separable metric space then there is a $E \subset [0,1]^{\mathbb{N}}$ such that $X$ is homeomorphic to $E$ ($E$ is a $G_\delta$ set - is the intersection of denumerable ...
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50 views

Connectedness of the Hausdorff distance.

Does anyone know a proof of connectedness of the Hausdorff distance? I mean a proof of the following: Theorem If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected. ...
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What is “approximate compactness”? What is an example of an approximately compact set?

I read this: A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a ...
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sequence of open sets

Find the sequence of open sets in $\Bbb{R}$ like $\{G_n\}$ such that $\Bbb{Z}=\bigcap _{n=1} ^{\infty}G_n$. I think an answer is this: $$G_n=\bigcup_{m=1} ...
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32 views

Does this implies that two metric spaces are Equivalent?

If two metrics $d_i$ on the same set $X$ have the same Cauchy sequences (ie. if a sequence is Cauchy for the first metric, it is also Cauchy for the other one and vice versa).Does this imply that the ...
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34 views

Proof that $\mathbb Q_p$ is unique up to unique isomorphism preserving the absolute values

On pages 58-59 of Gouvea's $p-$adic Numbers: An Introduction, he gives the following proof that the field $\mathbb Q_p$, constructed using equivalence classes of Cauchy sequences, is unique up to ...
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23 views

How to define metric in the Space of Holomorphic Functions?

I am looking for a proper way to define distane on the space of Holomorphic functions defined on a domain $D$.Does the Montel's Theorem (Given below from Stein's Book) helps to Characterize Compact ...
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Jacobian for bi-Lipschitz mappings between Hilbert cubes

It is well known that for every bi-Lipschitz function $f:M\to N$ between finite-dimensional smooth manifolds has a Jacobian, i.e. there exists $J_f:M\to \mathbb{R_{\geq 0}}$ such that for every ...
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34 views

Expanding circle endomorphisms

When I was going through Dynamical systems by Brin and Stuck, I came across this question. I am interested in characterising the expansiveness for invertible linear operator through which ...
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237 views

To understand some terminology of metric spaces

In the course of my study of metric spaces I've come across some terminology which I can't seem to understand completely. So, assuming $X=\mathbb{R}$, and $\mathbb{Q}\subset X$ is the set of the ...
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3answers
35 views

Do we need to have a subsequence such that $\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a normed space and $(x_n)_{n\in\mathbb{N}}\subseteq X$. Can we prove that there is a subsequence ...
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26 views

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?
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83 views

Sequences - Bolzano-Weierstrass theorem

Let $\mathbb{R}^\mathbb{N}$ be the $\mathbb{R}$-vector space of all real sequences and for $(x_n)_{n\in \mathbb{N}}\in \mathbb{R}^\mathbb{N}$ let $l^\infty :=${$x=(x_n)_{n\in \mathbb{N}}\in ...
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16 views

Let $X,Y$ be metric spaces , $f : X \rightarrow Y$ be a continuous function , $A$ be a bounded subset of $X$ and let $B =f(A)$.

Let $X,Y$ be metric spaces , $f : X \rightarrow Y$ be a continuous function , $A$ be a bounded subset of $X$ and let $B =f(A)$. Then $B$ is : $(i) $ bounded $(ii) $ bounded if $A$ is also closed ...
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34 views

How is $\|\cdot\|_1$ defined on a finite-dimensional real vector space?

Let $V$ a normed space over $\Bbb{R}$, and let $S$ be a finite dimensional subspace. I'm trying to show that $S$ is complete, I've already seen this question has ben made, but I have a precise doubt. ...
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64 views

Is a Banach space also a metric space?

Since a Banach space is a complete normed vector space and a norm always induces a metric, a Banach space must be a metric space, right? If so, why is a Banach space defined as a complete normed ...
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32 views

Euclidean distance. Subsets within subsets. (Metric spaces)

1: Let $M=R^2$ and $d: M\times M \to R,~d(x,y) = \begin{cases}\|x-y\|&\text{if y=t$\cdot$x for a}~ t\in R\\\|x\|+\|y\| &\text{otherwise}\end{cases}$ whereas $\|x\|$ is the ...
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36 views

Metric space - complete - discrete

a)Let $M\neq \emptyset$ be a set. Show that, $d(x,y):= \begin{cases}0&\text{if x=y}\\1&\text{otherwise}\end{cases}$ is a metric on $M$. This is a discrete metric. Formulate and ...
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27 views

Proving triangular inequality for any metric with $|a-b|$ rather than $|a+b|$ as part of it

If I am trying to prove triangular inequality for a metric with $|a-b|$ in it, and any other quality, to prove triangular inequality(for showing its a metric) do I prove: $$|a+b|\leq |a+c|+|b+c|$$ ...
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35 views

Compatibility of topologies and metrics on the Hilbert cube

Consider the Hilbert cube $Y = [0,1]^\mathbb{N}$. It is easy to define four classes of metrics on $Y$ for $\gamma>0$ and $\omega>1$: $$d^\gamma_{sup,pol}(x,y) = \sup_{k\geq 1} ...
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For compact $A$, $\inf\{\varrho(y,x) : y \in A\}=\varrho(a,x)$

I need help with prooving that if non empty $A$ $\subset(X,\varrho)$ is compact, then: $(\forall x \in X) (\exists a \in A) \inf\{\varrho(y,x) : y \in A\}=\varrho(a,x) $ I found this solution: ...
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36 views

3 homeomorphisms between spaces (2 with jungle metric)

I've been studying for my final exam from topology and I found such an exercise. Let $X=([0,1]\times\{0\})\cup \bigcup_{n=1}^{\infty}(\{\frac{1}{n}\}\times[0,\frac{1}{n}])$ Let ...
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91 views

Is this (countable) product space complete?

Let $((X_n, {\rm d}_n))_{n \geq 0}$ a sequence of complete metric spaces. Suppose that all the metrics are bounded by $1$. Consider $X = \prod_{n \geq 0}X_n$ with the metric given by: $${\rm ...
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99 views

Alternate proof for Arzela-Ascoli

Im trying to finish a beautiful excercise, which consist of giving an alternate proof for the following corollary of Arzela-Ascoli´s Theorem. Given $X,Y$ metric spaces, $X$ compact, $Y$ complete, and ...
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Does there exist a path connected metric space , in which at least one open ball is countable ?

Does there exist a path connected metric space with more than one point , in which at least one open ball is countable ?
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41 views

Completion of this metric space

Let $d'$ be a metric on $C^1([0,1])$ as follows: $$ d':\ C^{1}\left([0,1]\right)^{2}:\ |f(0)-g(0)| + \sup\left\{ |f'(x)-g'(x)| \mid x\in[0,1] \right\} $$ I've already managed to prove that this is ...
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Metric in $\mathbb{P}_2$

I have to prove that $\mathbb{P}_2$ with the function $\delta(P,Q)$ defined by "Sine of the angle between two vector in $\mathbb{R}^3$ such that they correspond respectively to P and Q" is effectively ...
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Is it mathematically correct to say that if the metric is flat/curved the *shortest* path is/not a Euclidean straight line?

Is it mathematically correct to say that if the metric is flat/curved the shortest path is/not a Euclidean straight line? I am still hesitant to make this claim, due to at least one counter example. ...
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Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$

Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$ Prove $(X,d)$ is a metric ...
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Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
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The set of irrational numbers is not a $F_{\sigma}$ set.

I want to proove that the set of irrational numbers is not a $F_{\sigma }$ set and also the set of rational numbers is not a $G_{\delta}$ set using Baire theorem. I started with saying that ...
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25 views

Complete product of metric spaces

Prove that: If a product $X\times Y$ of metric spaces $(X,\rho_X)$ and $(Y, \rho_Y)$ with metric $\rho((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ is complete, then metrics $\rho_X$ and ...
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18 views

Continuous function on interval, how do balls look.

Consider the metric space $C([a,b]),d_1$. $$ d_1:\ C([a,b])^2 \rightarrow \mathbb{R}:\ (f,g) \mapsto \int_a^b|f(x)-g(x)|\ dx$$ Is this metric space a normed vector space? How do open balls look? The ...
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11 views

Clarification of Sequential characterization of closedness of the set

I've been trying to understand $ \Leftarrow $ part of proof from link http://math.stackexchange.com/a/153372/240184 I dont understand why Hence the closure of F is a subset of F, whence they are ...
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43 views

How do you call a metric space with “continuous” points?

I have the impression that "continuous space" is not a mathematically precise concept (as opposed to continuous functions that can be defined under various contexts). However, I find I need to refer ...
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20 views

Nonincreasing and nondecreasing sequences in Hausdorff metric

For every metric space $(X,d)$ we have the Hausdorff metric space $(\mathcal{H}(X),H)$ that assosiates with it, where $\mathcal{H}(X)$ is the space of nonempty compact subsets of $X$ and $H$ is the ...
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Subsequence implicate bounded and closed set

I've been thinking about that problem for a long time, now it is right time to ask! Problem: Proof that if $ K \subset \mathbb{R}^{d} $ is such a set that every sequence with elements in $ K $ ...
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Are there infinitely many non equivalent metric spaces on certain sets (?)

Two metric spaces X and Y are called equivalent if: $d_X (x,x_n) \to 0 \Leftrightarrow d_Y (x,x_n) \to 0 $ with $ n \to \infty $ I wonder whether, if you took a certain set (for example a finite set, ...
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Question about Rudin's Functional Analysis Closed Graph Theorem

In page 51 of Rudin's Functional Analysis, the closed graph theorem is proven, which says that if you have a linear map between two F-spaces whose graph is closed in the product space, then the map is ...
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Calculate the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...
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All lines of $\mathbb R^3$ are isometric to $\mathbb R$

I have just started reading Metric Spaces by Michael Searcoid. The first Chapter states a result : Suppose $n \in \mathbb N~\forall~ i \in \mathbb N_n,(X_i,\tau_i) $ is a non empty metric space. ...
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48 views

The shortest path in a metric space with a given metric

My questions seem to be very basic and intuitively correct but I can't formally prove them. Before learning metric spaces, for $R^2$, we always define the distance between 2 points as $d_2 = ...
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70 views

Is there a name of such functions?

Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions: Is there ...
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25 views

Boundary and limit points

Suppose that $\Omega \in \mathbb{R}^n$. Prove that if $\vec{x} \notin \Omega$ and $\vec{x}$ is a boundary point of $\Omega$, then $\vec{x} $ is a limit point of $\Omega$. My try: $\vec{x}$ is a ...
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39 views

Prove that if $(f_n)_{n \geq 1} \to f$ uniformly then $f_n \to f \in B(X)$

Let $B(X)$ be the set of bounded functions from $X$ to $\mathbb{C}$ where $X$ is any metric space. Let $(f_n)_{n \geq 1} $ be a sequence in $B(X)$. Show that if $f_n \to f$ uniformly where ...
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5answers
172 views

A finite set is closed

Question: Prove that a finite subset in a metric space is closed. My proof-sketch: Let $A$ be finite set. Then $A=\{x_1, x_2,\dots, x_n\}.$ We know that $A$ has no limits points. What's next? ...
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4answers
49 views

Transferring the usual metric from $\mathbb{R}$ to $(0, 1)$ gives us a complete metric space on $(0, 1)$?

I'm watching this video here - https://www.youtube.com/watch?v=zcAvVTFUxS8 The lecturer says that $\mathbb{R}$ and $(0, 1)$ under the usual metric are homeomorphic yet $\mathbb{R}$ is complete and ...