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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How do i show that if every continuous function on $X$ is bounded, then $X$ is compact? [duplicate]

Let $(X,d)$ be a metric space. Assume every continuous function on $X$ is bounded. Prove that $X$ is compact. Well, i don't know which continuous function should i fix to start an ...
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343 views

Hausdorff and Fréchet distances

citation from wikipedia: It is possible for two curves to have small Hausdorff distance but large Fréchet distance Can anybody give me an example where this occurs? (sub-question: is it even ...
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29 views

Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.

Let $X=(0,\infty)$. Define two metrics on $X$ by $d_{1}(x,y)=|x-y|$ and $d_{2}(x,y)=|x-y|+|\frac{1}{x}-\frac{1}{y}|$ for all $x, y \in{X}$. Let $(x_{n})$ be a sequence in $X$ and $x\in{X}$. ...
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113 views

On the Gromov-Hausdorff distance

I'm working on my bachelor thesis, and I'm studying principally on two textbooks (Selected Topics on Analysis in Metric Spaces [1] by Luigi Ambrosio and Paolo Tilli and A Course in Metric Geometry [2] ...
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38 views

Rational vs real metric space

How to prevent, in a lesson that deals with basic mathematics, that we give two definitions of a metric ? Because there is one, which takes value in $\mathbb{Q}$, to build $\mathbb{R}$, that we do not ...
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34 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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42 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 in |SET| $(Y, d)$ in |MET|, $f_n$, $f$ is in $Y^X$. Then $f_n\to ...
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127 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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153 views

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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133 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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184 views

Show that $f(\bar A) \subset \overline{f(A)}$.

Let $X$ be a metric space, and $Y\subset X$ a subset. A point $x\in X$ is adherent to $Y$ if $B(x;r) \cap Y \neq \emptyset$ $\forall r > 0.$ The closure of $Y$ is then defined as $\bar Y := \{x\in ...
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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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76 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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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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69 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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76 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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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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31 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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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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59 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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42 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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202 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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166 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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369 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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54 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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135 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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51 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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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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46 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 ...
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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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101 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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252 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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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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100 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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51 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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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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72 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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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 ...
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41 views

Are these metrics complete?

Determine if these subsets of R are complete with the Euclidean metric? a) $[0,\infty)$ b) $(0,\infty)$ I know the definitions of completeness and I know the Euclidean metric, but don't know how to ...
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Prove that if $B(x,r)$ and $B(x',r')$ are disjoint $\Longleftrightarrow d(x,x') \ge r+r'$

Assuming that $d(x,x') \ge r+r'$, and proving that they are disjoint is easy. It's the other side that I'm having difficulty with. This seems like a really easy problem, but i'm having difficulty ...
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161 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
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Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
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60 views

(Proof Checking) Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$.

Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$. I am tempted to use the following argument. Let $U = \{U_i|i\in I\}$ be some open cover of $C\cap ...
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find a sequence of closed connected subsets $V_n$ of $\mathbb R^2$ s.t. $V_n\supseteq V_{n+1}$ and $\cap^{\infty}_{i=1}V_i$ is not connected

Find an example of a sequence $V_n$ of closed and connected subsets of the Euclidean plane satisfying $V_n \supseteq V_{n+1}$ such that $\cap_{i=1}^{\infty}V_i$ is not connected. Normally I would ...
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95 views

Hausdorff distance for empty sets?

The Hausdorff distance is defined for non-empty sets. What would be a reasonable generalization of the definition for the case when one of the sets is empty, if the generalized distance should remain ...
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40 views

Closure of a particular set, $c_{00}$.

I have been learning the concept of closure in metric spaces and all has made sense so far. However, I have come across a particular example that is troubling me in terms of extracting the possible ...
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53 views

Closure of interior proper subset of interior of closure

$\newcommand{\intr}{\mathrm{int}}$ I need to give an example of a metric space $(X,d)$ and $A ⊆ X$ so that $\overline{\intr(A)} ⊂ \intr (\overline{A})$, where $\overline{B}$ refers to the closure of ...
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69 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
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70 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...