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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Munkres Section 20 Exercise 3b. Proof verification.

Let $X$ be a metric space with metric $d$. (a) Show that $d:X\times X\to\Bbb R$ is continuous. I've shown this already. (b) Let $X'$ denote a space having the same underlying set as $X$. Show that ...
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2answers
29 views

Prove $\tilde d: M\to \mathbb{R}; x\mapsto \tilde d(x) = d(x,A)$ is continous

Let $A$ be a non-empty subset of a certain metric space $M$. Prove that $\tilde d: M\to \mathbb{R}; x\mapsto \tilde d(x) = d(x,A)$ is continous. (where $d(x,a) = \inf\{d(x,a): a\in A\}$) ...
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2answers
56 views

Is it always true that the complement of a closed set is open?

In a metric space $M$, by definition, a subset $F$ is closed if $M\setminus F$ is open. However in a general topological space $T$, say with topology $\mathcal{T}$ is this always true? For example ...
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3answers
52 views

Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$

Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$ I have encountered this exercise and I don't know where to start. There is a lot of important ...
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8 views

How to obtain the metric tensor of a principal bundle total space given a connection (assuming it is metric compatible) in the total space?

The title says it all. In a principal bundle I know the connection defined in the total space. How can I calculate the metric that would be compatible with this connection.
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1answer
25 views

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete.

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete. I've already seen that this question has been asked, and already answered, however, I've managed ...
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0answers
23 views

Prove that $X$ contains exactly two clopen sets if and only if every nonempty proper subset of $X$ has a nonempty boundary.

Let $X$ be a metric space. Prove that $X$ is connected if and only if every nonempty proper subset of $X$ has a nonempty boundary. proof: Suppose X is connected, then $X$ contains exactly two ...
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1answer
42 views

Show that $a\to f(a)$ from $A$ to $S$ is continuous.

I am reading "Continuity" in Metric Spaces Suppose $S\subset \mathbb R$ is a closed set. Suppose $A\subset \mathbb R$ has the property that for every $a\in A$ there is a unique nearest point $f(a)$ ...
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1answer
70 views

uniformly bounded equicontinuous family of real valued functions defined on a metric space [closed]

I'm new to Stack. Got an exam soon and wonder if anyone could help me with two questions: $1$) Let $\mathcal{F}$ be a uniformly bounded equicontinuous family of real valued functions defined on ...
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33 views

Contracting subsets

Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
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1answer
35 views

Metric spaces and fix point [closed]

I saw this problem in my course of Intr. to the topology: Let $(X,d)$ be a compact metric space and $$f :(X,d) \rightarrow (X,d)$$ a continuous function such that: $\quad d(f(x);f(y)) < ...
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2answers
40 views

Show that $B(X)=\{f:X \rightarrow \mathbb{R}: f \text{ is bounded}\}$ is complete.

Given a metric space $(X,d)$ consider the metric space $B(X)=\{f:X \rightarrow \mathbb{R}: f \text{ is bounded}\}$ with the distance $d_{\infty}(f,g)=sup_{x\in X}|f(x)-g(x)|$. Show that ...
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22 views

Is this proof involving complete metric spaces correct?

Show that if every closed ball of a metric space $(X, d)$ is complete then $ X$ is complete. I thought the following: given $(x_n)$ a Cauchy sequence in $X$, we have that the set $A= \{x_{1}, ...
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1answer
6 views

2 dimensional representations of v dimensional points based on distance

I have a set of points $x_1, \dots, x_n \in \mathbb R^v$ I have a measure of the distance between each one of these points $D \in \mathbb R^{n\times n}$ where $D_{i,j}= distance(x_i, x_j)$ I would ...
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1answer
35 views

$[0,1]\times\mathbb{N}/(0,k)$ not metrizable.

$X = [0,1]\times\mathbb{N}/((0,1)\sim(0,2)\sim\dots)$. I read that $X$ is not metrizable since sequence $\{(\frac{1}{n},n)\}$ is closed in $X$ and therefore does'n have limit. But i don't understand ...
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1answer
63 views

Some questions about écarts

An écart for a set $X$ is a non-negative real-valued function $e:X\times X\rightarrow \mathbb{R}$ such that $e(x,y)=0$ if and only if $x=y$; for each positive number $s$ there is a ...
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1answer
53 views

multiplying two metrics

Let $(X,d)$ and $(X,d^\prime)$ are metric spaces ,is $d×d^\prime$ metric on $X$ ?I try to prove triangle inequality , I write two triangle inequalities for $d $ and $ d^\prime$ but it not true.
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1answer
74 views

Proving $\ell^p$ is complete

Let be $1\leq p\in\mathbb{R}$, denote: $$\ell^p(\mathbb {R})=\left\{(x_n)\subset \mathbb{R}: (x_n) \mbox{ is a sequence with } \displaystyle\sum_{n=1}^{\infty}|x_n|^p<\infty \right\}$$ ...
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1answer
134 views

Is this a compact space?

Let $A=\{x:d_\infty(x,0)\le 1 \}$, the subspace of the space of bounded sequences $x=(x_n)^\infty_{n=1}$, $x_n\in \mathbb{R}$, with metric $\{x:d_\infty(x,y)= sup_n |x_n-y_n| \}$. The answer says it ...
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3answers
37 views

Conceptual problem regarding distance between two sets.

Given a metric space $(X,d)$ and two non empty subsets $A,B \subset X$ we define the distance between $A$ and $B$ as $$ d(A,B) = \inf\, \{d(a,b) : a\in A, b\in B\} $$ My question is the following: ...
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39 views

Is the distance function continuous? [closed]

Is the distance function continuous? I know that distance function is continuous, give an example of distance function that is not continuous.
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54 views

Example 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be open?

Let $N$ be the following subset of $\mathbb{R}^2$: $$N \colon= \{ \ (x,y) \in \mathbb{R}^2 \ \colon \ \vert x \vert < \frac{1}{y^2+1} \ \}.$$ Then intuitively it is apparent that $N$ is open. ...
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28 views

Characterization of a quotient space

Given the space $C^n[0,1]$ of all real functions of class $C^n$ in $[0,1]$, let $\tilde{d}^j := d_\infty(f^{(j)},g^{(j)})$ a pseudometric $(j=1,\dots,n)$ on $C^n[0,1]$. Here $f^{(j)}$ mean the ...
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1answer
35 views

Prove/disprove space is complete with metric defined by an integral (triangle inequality still missing in metric part)

I have a two part question: I need to show that $d(f,g)=\int_{-1}^1\! |f(x)-g(x)| \, \mathrm{d}x$ is a metric in $C((-1,1),\mathbb{R)}$ and furthermore prove/disprove that the space ...
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1answer
23 views

Equivalence of sequence spaces

Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i ...
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13 views

find mean of matrices $A_i, A_j$ given $d_{A_{ji}}=\ln{\left|\left| A_{ji} \right|\right| \left|\left| A_{ji}^{-1} \right|\right|}$

Given a finite set $\mathbb{A}$ of $k$ like-shaped, square, non-singular matrices $A_i\in\mathbb{R}^{n\times n}$, let's define $A_{ji}=A_j A_i^{-1}$, then the distance of the two matrices $A_i, A_j$ ...
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16 views

Understanding extremal Lipschitz functions

I am new to concept of extremal Lipschitz functions and I have several basic question I'm still unsure about. To fix notation let $(X,d)$ be a metric space, $Lip(X)$ Banach space of Lipschitz ...
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39 views

Prove that if $S\subset \mathbb{R}^n$ is not countable, then there exists $x \in S$ such that $x$ is a condensation point.

Let $S \subset \mathbb{R}^n$ with the usual metric. A point $x \in \mathbb{R}^n$ is said to be a condensation point of $S$ if for all $r>0$, $B(x,r)\cap S$ is not countable. Show that if $S$ is ...
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27 views

trouble in getting triangle inequality

Let $l_{2}$ be the set of all infinite sequences , $ (x_{n})$ such that $\sum_{n=1}^ {\infty} x_{n}$ converges. Define $$d(x,y)= \sqrt{\sum_{n=1}^{\infty} (x_{n}-y_{n})^{2}}$$ for each $x=(x_{n})$ ...
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2answers
34 views

Metrics on the set of natural numbers

I am trying to find a metric d on $ \mathbb{N} $ that is not equivalent to the discrete metric $ d_{\{0,1\}} $. Thus far I got a metric with the following properties: $ d(x, x_n) \in [0,1) \forall n ...
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32 views

Covering number definition, general metric space question.

While reading "On the mathematical foundations of learning" by F.Cucker and S.Smale I came across this definition: Let $S$ be a metric space and $s>0$. We define the covering number ...
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Topologically equivalent metric

Show that in $\mathbb{R}$ the distance $d'(x,y)=\left|\frac{x}{1+|x|} - \frac{y}{1+|y|} \right|$ is topologically equivalent to the usual metric in $\mathbb{R}$, $d(x,y)=|x-y|$ But ...
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every nonempty compact, locally path-connected and connected metric space is path-connected [duplicate]

I wanna prove that if $M$ is nonempty compact, locally path-connected and connected metric space then it is path connected. I think to prove this the best way is to show that between every to points ...
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2answers
75 views

Prove that $f^{-1} (F)$ is closed

A set $F \subset \mathbb R$ is closed if for any convergent sequence $\{x_n\}$ in F converges, we have $\lim_{n \to \infty} x_n=x \in F $. How to Prove that if $f :\mathbb R \to \mathbb R$ is ...
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1answer
33 views

Why is the open interval $(0, 1)$ a Polish space?

Wikipedia gives as an example for Polish spaces the open interval $(0, 1)$. Can somebody explain to me how $(0,1)$ can be Polish? $(0, 1)$ has to be metrizable so that it is complete, which means ...
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5answers
358 views

Express unit sphere as countable union of great circles?

Let $S = \{x\in \mathbb{R^3} | d(x,(0,0,0))=1\}.$ Is it possible that $S$ is a countable union of “great circles”? A great circle is the intersection of $S$ with a plane through $(0,0,0)$. What ...
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1answer
44 views

Prove that totally bounded metric space is separable

I'm reading the proof that says every totally bounded metric space is separable. The proof goes like this: Let $n$ be a positive integer. Then there exists $x_{n1},...x_{nm}$ such that the open ...
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34 views

Typical example of topology of metric space(NBHM)

State whether the following subsets of $M_2(\mathbb{R})$ (with standard topology) are open, closed or neither open nor closed. (a) The set of all matrices in $M_2(\mathbb{R})$ such that neither ...
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20 views

homeomorphism inbetween the $\mathbb{R}^n$ and an open unit cube

I want to find a homeomorphism that maps the open cube $W = (-1,1)^n\subseteq \mathbb{R}^n$ to the $\mathbb{R}^n$. I know that these two are homeomorphic, but I don't know where to start when it ...
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18 views

incomplete vector space of continously differentiable functions

Consider the vector space $C^1[a, b] := \{f: [a, b] \to \mathbb{C} \space |\space f$ continuously differentiable$\}$. I now want to show that ($C^1[-1, 1]$, $||.||_\infty)$ is not complete (using ...
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Can a metric subspace be completely covered by balls after a finite number of steps?

Let $X$ me a metric space with distance $d$ and $A$ be a subspace of $X$. Let $B_\varepsilon(x)$ be the open ball centered in $x$ with radius $\varepsilon$, i.e. $\{y\in X\mid d(x,y) < ...
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1answer
93 views

Greatest Lower Bound of the Set of Upper Bounds for a Function

I'm currently in the process of reading a paper, and am trying to work through some of the details on my own. In order to ask my question more effectively, I'm going to begin with a little background: ...
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1answer
24 views

On general topological spaces and $C(X, \mathbb R)$ , where for closed sets $A,B$ in $X$ , $I_A=I_B \implies A=B$

Let $X$ be a metric space and $C(X, \mathbb R)$ be the ring of all real valued continuous functions from $X$ . For $A \subseteq X$ , let us define $I_A :=\{f \in C(X, \mathbb R) : f(x)=0 , \forall x ...
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40 views

The set of points of continuity of a real-valued function on a metric space is a $G_\delta$ set

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. I know lots of other ...
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57 views

The distance between two sets does not change if closure is taken

Given $ (X, d)$ a metric space, $ A, B \subset X$, show that $ d(A, B)=d (\overline {A}, B) $. I'm not being able to show that $ d(A,B) \leq d (\overline {A}, B) $. Can anybody help me? The set ...
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1answer
30 views

Prove that a function is open

Let $X,Y$ metric spaces and $U \subset X , V\subset Y$ open sets. Let $f:U\rightarrow V$ be a homeomorphism. Prove that $f$ is an open map. I need to show that for every open subset of $U′⊂U$, $f(U′)$ ...
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40 views

How is the convergent sequence $\frac{1}{n-1}$ bounded?

In a metric space all convergent sequences are bounded. This example in the real numbers should then be bounded but, it is infinite at n=1 so I do not understand how this can be true. In the proof ...
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1answer
31 views

Prove that if $A \cap B = \emptyset$ and if dist($A,B$) = inf { $\rho (x,y) : x \in A$ and $ y \in B$ } then dist($A,B) > 0$.

Suppose that $X$ satisfies the Bolzano- Weierstrass Property and that $A$ and $B$ are compact subsets of $X$. Prove that if $A \cap B = \emptyset$ and if dist($A,B$) = inf { $\rho (x,y) : x \in A$ ...
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1answer
24 views

“distance” metric between two bases modulo determinant, rotation and chirality

I'd like some kind of metric that tells me how similar two complete, not necessarily orthonormal bases (represented by non-singular matrices $B_1, B_2 \in \mathbb{R}^{n \times n}$) are to each other, ...
3
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1answer
28 views

Frechet metric: troubles understanding $d(x^{(j)},0)\to0\iff x_i^{(j)}\to0$ $\forall i\in\mathbb N$

Consider the metric $$ d(x,y)=\sum_{k=1}^\infty\frac1{2^k}\frac{|x_k-y_k|}{1+|x_k-y_k|} $$on $\mathbb R^{\mathbb N}$ with $x=(x_k)$ and $y=(y_k)$. Let $x^{(j)}\in\mathbb R^{\mathbb N}$ for all ...