# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...