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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135 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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1answer
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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1answer
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
36 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 ...
2
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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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0answers
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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0answers
18 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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1answer
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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1answer
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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1answer
33 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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3answers
32 views

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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0answers
18 views

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
360 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
45 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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0answers
36 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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1answer
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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1answer
20 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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74 views

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
94 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
25 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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1answer
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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2answers
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
33 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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1answer
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 { $\rho (x,y) : x \in A$ ...
0
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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 ...
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1answer
45 views

Fractal dimension of a dense subset

Let $M$ be a metric space and $S\subset M$ a dense subset. For vague reasons (below), it seems to me that the upper box-counting dimension of $S$ should be equal to that of $M$, but I don't quite see ...
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2answers
50 views

Help with proving $f : X \rightarrow \mathbb{R}$

Let $X$ be a metric space with distance $d$ and let $p \in X$ be fixed. Define $f(x) = d(x,p)$. Prove that $f: X \rightarrow \mathbb{R}$ is a continuous function.
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1answer
30 views

Horosphere of a metric tree

I have a project I have to do. In order to do it I need to investigate W.E. Grosso's translation of "The Green Book" on Hyperbolic Group Theory, as found here. I try to understand the term horosphere ...
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0answers
17 views

When is 1-Lipschitz function a distance?

Let $(X,d)$ be a metric space and $f:X\rightarrow \mathbb{R}$ a 1-Lipschitz function. Are there some criteria for $f$ being of the form $f(x)=\pm d(x,x_0)+c$?
2
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1answer
74 views

Complete metric space of sequence of positive integers [duplicate]

Let $(A,d)$ be the space $\mathbb{N}^{\mathbb{N}}$ of sequences of positive integers where $d((a_i)_i, (b_i)_i)= \frac{1}{n}$ where $n$ is the least coordinate at which $(x_i)_i$ and $(y_i)_i$ ...
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1answer
40 views

Metrizability of a topological space

If we have a topological space $(X,\mathcal{T})$ and a metric $d$ on $X$ s.t. for any sequence $(x_n)_n$ convergence of the sequence $x_n$ to some $x \in X$ for the topology $\mathcal{T}$ is ...
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0answers
43 views

How it can be possible? [duplicate]

I can't understand how it can be possible? Prove that the interval $[0,1]$ has a uncountable partition $\mathcal P$ such that each $D\in \mathcal P$ is uncountable and dense in $[0,1]$.
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3answers
41 views

Closed subset of the $\mathbb{R}^n$

I want to show that $U = \{(x, y) \in \mathbb{R}^2|xy ≤ 1\}$ is a closed subset of $\mathbb{R}^2$. Yes there are (easy) ways to do this using functions, but what's the (easiest) way to prove this ...
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2answers
19 views

Separable metric space and discontinuouty point of the function defined on it

I'm trying to show the following: Let $(X,d)$ be a separable metric space and $f:X\to \mathbb R$ be a function such that, for every point $x\in X$, $\,\,\,$ $lim_{t\to x}f(t)$ exist. consider: ...
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1answer
19 views

Show that $D(A,B):=\inf\{r:A\subseteq N_r(B) \text{and}B\subseteq N_r(A)\}$ is a metric on bounded closed subsets of $X$.

Let $A,B$ be non empty bounded closed subsets of metric space $(X,d)$, define $D(A,B):=\inf\{r:A\subseteq N_r(B) \text{and}B\subseteq N_r(A)\}$. Show that $D$ is a metric on bounded closed subsets ...
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0answers
8 views

Is the mapping achieved by the discrete Frechet Distance the best achievable pair decomposition?

I have the following proble: There are 2 sequences $P$ and $Q$ composed by $M$ and $N$ points respectively. I want to calculate the discrete Frechet distance between them. This operation outputs also ...
2
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1answer
47 views

If $X$ contains a countable dense subset, then $A$ denumerable or finite.

Suppose that $\{ V_{\alpha}\}_{\alpha \in A}$ is a collection of nonempty open sets in $X$ which satisfies $V_{\alpha} \cap V_{\beta} = \emptyset $ for all $\alpha \neq \beta$ in $A$. Prove that if ...
2
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1answer
24 views

Identify which of the following sets are compact and which are not.

Identify which of the following sets are compact and which are not. If $E$ is not compact, find the smallest compact set $H$ (if there is one) such that $E \subset H$. a) $E = \{ \frac{1}{k} : k \in ...
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1answer
41 views

Hausdorff distance

Let $A=\{(x,y)∈R^2: x^2+y^2\le4\}$ and $B=\{(0,y)∈R^2:|y|\le3\}$. Determine Hausdorff distance between $A$ and $B$. I wrote $d(2)(B,A)=((0,3),A)=((0,3)(0,2))=1$. What about $d(A,B)$? ...
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3answers
24 views

characterisation of a dense subset in a metric space

Let $(X, d)$ be a metric space. Using only the definition that $M \subseteq X$ is dense in X if $\overline{M} = X$, I want to show for any subset $M \subseteq X$ that: M is dense in X ...
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vote
2answers
36 views

open subset of $\mathbb{R}^2$

If we consider $M \subseteq \mathbb{R}^2$, where $$M = \{(x, y) \in (0, \infty) \times (0, \infty)| \space x y < 1\}$$ I want to prove that M is an open subset of $\mathbb{R}^2$. Now it may seem ...
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1answer
33 views

Differential operator is not continuous between this metric spaces

Let $\mathbb{D}$ be the set of functions $f:[0,1]\to \mathbb{R}$ of class $C^1$ (differentiable with continuous derivative). Let $\mathcal{C}[0,1]$ be the set of continuous functions in $[0,1]$ ($\to ...
0
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2answers
33 views

Difference between open sets and open balls in metric space

Let $X$ be a separable metric space and let $\mathfrak{M}$ be the $\sigma$-algebra generated by open balls in $X$. Show that $\mathfrak{M}$ contains all the open sets in $X$ and all the closed ...
0
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3answers
50 views

Show that this metric is not complete

Show that the set of continuous functions $C[a,b]$ under the metric $\rho(f,g)=\displaystyle{\sqrt{\int_a^b|f(t)-g(t)|^2dt}}$ is not a complete metric space for $f,g\in C[a,b]$ and $t\in[a,b]$. To ...