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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Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
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317 views

Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?

Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb ...
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20 views

Is there a distance metric for dot product similarity that preserves the ordering of nearest neighbors?

The dot product and cosine similarity measures on vector space are frequently used in machine learning methods. However, efficient data structures and algorithms often require a metric space distance ...
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1answer
20 views

Finite set in a metric space [on hold]

Let $X={a,b,c,d}$ and $d$ is trivial. Task Is $A=\{a,b\}$ open or closed? My approach I figured both: since $A$ is an interior to $B(a,1/2)\cup B(b,1/2)$, it is open, and since the complement of ...
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35 views

Quasi-isometry of Schreier coset graphs

Let $G$ be a finitely generated group and $H$ a subgroup. For any choice of a generating set $X$ we can form the Schreier coset graph. Is this graph independent of the generating set up to ...
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15 views

To show closedness of a subset in a metric spaces

Let $(X, d)$ be a metric space and $p\in X$, $\delta>0$ be fixed. Let $$A=\{q \in X : p \in X, d(p,q)>\delta\}$$ How to show that $A$ is closed? I tried to show that directly by taking $A$'s ...
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23 views

A lemma on function spaces

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me? Lemma: let $X$ be a set $(Y, d)$ in the metric space, $f_n$, $f$ is in $Y^X$. Then ...
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1answer
227 views

Showing that $d(m,n)=|m^{-1}-n^{-1}|$ is not a complete metric on $\mathbb Z^+$

Let $X$ be a set of all positive integers and define metric $d$ on $X$ by $d(m,n)=|m^{-1} - n^{-1}|$. I'm required to show $(X,d)$ is not a complete space. SOLUTION: Let $\{x_n\}$ be any Cauchy ...
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229 views

Proving a continuous function $f:X\to Y$ is uniform continuous if $X$ is compact.

I'm reading the proof of "if there's a continuous function $f:X\to Y$ where $X$ is a compact metric space and $Y$ is a metric space, then $f$ is uniformly continuous on $X$." The proof proceeds thus: ...
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88 views

Determine whether this d is a metric?

Suppose we have a function $d:\mathbb{Z}^{2}\times\mathbb{Z}^{2}\rightarrow\mathbb{R}$ where $\forall(x_{0},y_{0}),(x_{1},y_{1})\in \mathbb{Z}^{2}$, we have.. $d\left( \left( x_{0},y_{0} \right), ...
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16 views

Describe the Unit Ball

I was asked to describe the unit ball in $C(I)$. All I could come up with was that by definition $B_{1}(0):=\{x \in C(I) : ||x||_{\infty}<1 \}$, where $||x||_{\infty}:=\sup_{t \in I} |x(t)|$. Thus, ...
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1answer
24 views

Is the set of all integers with metric $d(m,n)=|m-n|$ a complete space?

Consider the set of integers with a metric defined by $d(m,n)=|m-n|$.Is this set complete with respect to this metric? If it is a metric, then I am stuck here. How can a Cauchy sequence have a limit ...
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19 views

Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
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1answer
24 views

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact.

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact. Some helpful definitions: bounded - A subset $S$ of a ...
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48 views

basic question of topology involving compactness and convexity

Consider in $R^n$ a compact and convex set $A$ with $int(A) $ nonempty. then $\overline{int(A)} = A$ ?. i have no idea to prove this. In this direction i only know the following (and hard to ...
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24 views

define distance in a manifold over the reals

G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, ...
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205 views

Discrete metric space [on hold]

Let (X,d) be a discrete metric space. For any A that is a subset of X, is the following true: The closure of A is the the set {{0} union {1}} because that is the set with all of the limit points. The ...
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21 views

Is a closure a disjoint union of limit points and isolated points

Definition) A point $x\in X$ is a limit point of S if every ball $B(x;r)$ contains infinitely many points from $S$. A point $x\in X$ is called an isolated point of S if $\exists r > 0$ such that ...
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19 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
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Product topology and standard euclidean topology over $\mathbb{R}^n$ are equivalent

I would like to know why the product topology and the standard euclidean topology over $\mathbb{R}^n$ are equivalent. I already found the question here: Showing that the product and metric topology ...
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39 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
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What's the motivation behind metric spaces?

So a metric space is a collection of points together with operations, and where we can determine the distance between any of these points. And it must satisfy 4 axioms which are: For all x in that ...
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11 views

Which Similarity Metric should I use?

I have currently created three similarity metrics, but the point is I have no idea which one I should use and when one is better for use than the other. I am currently using these metrics for people ...
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20 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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164 views

Triangle Inequality on a different normed space

Let $x=(x_1,x_2)$, the norm is given by $[x]=\sqrt{x_1^2+x_1x_2+x_2^2}$ I need to show the triangle inequality holds. So $y=(y_1,y_2)$ and from $[x+y]\le[x]+[y]$ I got ...
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20 views

What does the sup function mean in the context of metrics for probability measures/distances/differences?

I was studying different probability metrics and distances and came across the following source: ...
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54 views

Using the topology of uniform convergence for functions over non-compact spaces

Let $(X, d)$ be a (complete) metric space, and $C(X)$ be the space of continuous maps over $X$. If $X$ is compact, one often uses the topology of uniform convergence when analyzing $C(X)$. If $X$ is ...
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52 views

Is the following statement true?

Let $(X_1, d_1),\ldots,(X_n, d_n )$ be metric spaces, $ X: = X_1 \times \cdots\times X_n$ be their Cartesian product with metric $d$. Let $ \pi_i : X \to X_i$ be the projection for $ 1 \le i \le n$. ...
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66 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
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1answer
35 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
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37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
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296 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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1answer
18 views

continuous functions on metric space

Assume $f:X\rightarrow Y$, where $X$ and $Y$ are two metric spaces. If $f(\overline{E})\subset \overline{f(E)}, \, \forall E\subset X$, then how can we prove that $f$ is continuous? Thank you for ...
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22 views

A complex metric

Given the following definition $d(z , w) = \begin{cases}0 & z=w \\ |z|+ |w| & z\neq w \end{cases}$ I have to prove that $d(z,w)= 0\Rightarrow z = w$ Which is in part of checking that $d$ is ...
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1answer
29 views

A simple question on Hausdorff distance

Let $(A_n)$ be a sequence of compact sets in $R^n$ and consider $K$ and $A$ compact sets in $R^n$. Suppose that $A_n \cup K \rightarrow A \cup K$ in the Hausdorff distance. Then $$ A_n ...
2
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1answer
63 views

Cauchyness vs. Double Limits

Maybe there are some textbooks which might treat cauchyness by taking double limits... My question: Is it sufficient and necessary to consider the double limit: $$x_n\quad \text{cauchy}\quad ...
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3answers
38 views

Double Limit implies Successive Limits

I know it seems very stupid question, but is it right that: Suppose $X$ being a complete metric space. Then: $$\lim_{(m,n)}x_{(m,n)}\quad\text{exists} \quad\Rightarrow\quad \lim_n\lim_m ...
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13 views

Verify global Lipschitz condition

Consider the realized observations $z_1,...z_i,...,z_n$ i.i.d. Let $$ \hat{Q}_n(\theta)=\frac{1}{n} \sum_{i=1}^{n} 1\{z_i\in S(\theta)\} $$ I have to verify the following global Lipschitz condition: ...
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1answer
27 views

The Open Set $X-\lbrace x \rbrace$

I am task with proving the following: if $x \in X$ then $X- \lbrace x \rbrace $ is an open set I kind of have an idea but I am unsure about it and how to express it. I was thinking about using the ...
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864 views

An inequality for metric spaces: $|d(x, z) − d(y, z)| \le d(x,y)$

Question : Prove $|d(x, z) − d(y, z)|$ is less than or equal to $d(x, y)$. I know I have to use the triangle inequality but I'm just not sure how to apply it with a negative $d(y,x)$.
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How to show that $(C^0((a,b)), d_\infty)$ is not a metric space

Let $d_\infty:C^0([a,b]) \times C^0([a,b]) \to [0,\infty)$ be defined as $$ d_\infty(f,g)=\sup\limits_{x \in [a,b]} \left\{ |f(x) - g(x)| \right\} $$ I have already shown that $(C^0([a,b]), ...
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1answer
40 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
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1answer
30 views

Show that the interior of the set A is empty?

Consider $A = \{(x, \sin\frac{1}{x}) \mid 0< x \leq 1 \}$, a subset of $\mathbb R^2$. Find int($A$). We can see graphically that the interior of $A$ is definitely empty, but I want to check by the ...
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1answer
61 views

When is distance to the boundary always less than that to the exterior?

Let $X$ be a metric space with the distance function $d$. Given a subset $S \subseteq X$, what are the required conditions on $X$ and $S$ are so that $d(x, \partial S) \leq d(x, \operatorname{ext} S)$ ...
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1answer
28 views

Prove that metric space is complete

I have metric space: $$ X = <[0,+\infty), \rho>, \rho(x,y) = |ln(1+x) - ln(1+y)|$$ I know it is complete, but I don't know how to prove it. How can I prove that fact?
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209 views

Not Complete Metric Space

Suppose I have a metric $\rho: R^n$ x $R^n \rightarrow R$ defined by $\rho (x, y) = \frac{d(x, y)}{1+d(x, y)}$ and I need to show that $(R^n, \rho)$ is not a complete metric space? Can anyone suggest ...
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1answer
44 views

Set theory: not really understanding what the question is asking..

Here is the problem. Let $M$ be the metric space of all real numbers, and let $x_0 \in M$. Prove that there exist exactly two isometries of $M$ that leave $x_0$ fixed. I am having trouble ...
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1answer
37 views

Proving the distance function $|f - g|_u = \sup \{ |f(x) - g(x)|: x \in S \}$ defines a metric space

Let $S$ be a closed and bounded subset of $\mathbb{R}$. Define the "functional distance" between $f$ and $g$, both functions from $S$ to $\mathbb{R}$, to be $$ |f - g|_u = \sup \{ |f(x) - g(x)|: x ...
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3answers
344 views

Proving that if $X$ is a complete metric space and $A\subset X$ is nowhere dense in $X$, then there is an open set in $X$ disjoint with $A$.

Let $A\subset X$ be a nowhere dense set, where $X$ is a complete metric space. My book says there is an open set $S$ of radius less than $1$ such that $S$ is disjoint with $A$. I'm confused as to ...
0
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2answers
23 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...