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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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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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
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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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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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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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, ...
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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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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
29 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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16 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 $f(x)=\pm d(x,x_0)+c$?
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1answer
73 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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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
18 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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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 ...
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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 ...
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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
40 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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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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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
32 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 ...
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32 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 ...
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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 ...
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1answer
40 views

Proper Proof for Completeness of $\mathbb{R}$ with the Euclidean Metric

My code can't be uploaded because it doesn't work with the websites coding, but here is a pdf of my LaTex code. My question is, is this a proper proof? It feels as if I'm missing something important. ...
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2answers
25 views

metric on the set of complex sequences

Let X be the set of complex sequences $(a_n)_{n\in\mathbb{N}}\in \mathbb{C}$. Show that the transformation: $$ d((a_n), (b_n)) := \sum_{n=0}^\infty \frac{1}{2^{n+1}} \frac{|a_n - b_n|}{1 + |a_n - ...
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1answer
22 views

equations for an open ball in a normed space

Let $(V, \|\cdot\|)$ be a normed space. Show that for an open ball $B_1(0) \subseteq V$, it holds true that: $∂B_1(0) = \{x \in V: d(x, 0) = 1\}$ where $d(x, 0) = \|x\|$. Also, figure out the ...
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59 views

Is this an existing matrix distance/metric?

I was thinking about comparing different basis transformations and came up with this distance function: $$d(A,B)= \dfrac{||A - B||^2}{||A|| + ||B||}$$ I am using the Schatten-1-norm as the norm here ...
3
votes
1answer
30 views

Difference between continuous and uniformly continuous functions on a dense metric subspace.

Let $X$ be a dense subset of metric space $(\tilde X,d)$. Let be $(Y,d')$ be a complete metric space and $ f: X \rightarrow Y$ a continuous mapping. It follows from density that for all points in ...
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the examples of subspace embedding which are not Oblivious

For the definitions of Oblivious Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf. Then, can any one show the examples of subspace embedding which are ...
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0answers
29 views

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$? [duplicate]

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$ for the metric induced from the Euclidean metric on $\mathbb{R}$? I think that yes, it is nowehere dense because ...
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14 views

the difference and similarity between 'subspace embedding' and ' dimension reduction'

Can someone show me the difference and similarity between 'subspace embedding' and 'dimension reduction' using the mathematical definition? Thanks a lot.
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44 views

When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
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On a question about finite metric spaces

Let $(X,d)$ be a metric space such that every continuous function $f:X\to \mathbb R$ has a finite Image. prove that, $X$ is finite. I tried this: Let $x_0$ be arbitrary element of $X$ and define: ...
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27 views

Infinite Intersection of Open Sets [closed]

Give an example of an infinite intersection of open sets which is not open.
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1answer
30 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
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1answer
42 views

Find two disjoint open sets $U, V$ such that $A\subseteq U, B\subseteq V$ where $A,B$ are closed.

Let $A, B$ be two disjoint closed subsets of a certain metric space $(M,d)$. Show that there exist disjoint open subsets $U, V \subseteq M$ such that $A\subseteq U, B\subseteq V$. Give ...
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41 views

Metric space analog of the definition of continuity

The definition of continuity in topological spaces is given as: The function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at the point a in $\mathbb{R}^n $ iff given any open ball ...
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34 views

constructing an example of a ball of larger radius, contained in a ball of smaller radius

I've found an example where there is a ball of larger radius contained in a ball of smaller radius, but I'm not sure how it works: Let $X = \{ x \in \mathbb{R}^2 : x_1^2 + x_2^2 \leq 9 \}$ with the ...
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1answer
27 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?

We call two metrices equivalent if for all sequences $x_n,y_n\in\mathbb R$ it holds $\lim_{n\to\infty}d_1(x_n,y_n)=0 \iff\lim_{n\to\infty}d_2(x_n,y_n)=0$ . I have given $d_1(x,y)=|x-y|$ and ...
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2answers
23 views

Show that any subset of $(\mathbb{N},d)$ is open and closed

Show that any subset of $(\mathbb{N},d)$ is open and closed, where $$d(m,n) = \frac{|m-n|}{1+|m-n|}$$ my attempt: let $A \subset \mathbb{N}$ then for any $x \in A$ we have that $B(x,1/3) = \{x\} ...
5
votes
1answer
38 views

If $E \subset\mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E) \in E$

Suppose that $E \subset \mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E)\in E$. attempt: Suppose $E$ is compact, then $E$ is closed and bounded. Thus $\sup(E)$ and $\inf (E)$ exist. ...
2
votes
2answers
67 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
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3answers
32 views

Set that is bounded but not totally bounded: Reading textbook

I've been reading a Real Analysis textbook that my friend loaned to me. I have come across a proposition that says that a totally bounded set is bounded, but a bounded set is not always totally ...
4
votes
1answer
34 views

Prove there exists $a \in E$ such that $a = f(a)$, assuming $d(f(x), f(y)) \le Kd(x,y)$ with $K<1$

Let $f: E \rightarrow E$, $E$ a complete metric space. Assume that there exists $K$ such that $0 < K < 1$ and $d(f(x), f(y)) \le Kd(x,y)$ for all $x,y \in E$. Prove that there exists $a \in E$ ...