# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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### Show that $U \subset S$ open in S $\iff$ there exists an open set $U'$ in $X$ such that $U=U' \cap S$

Let X be a metric space and let $S\subset X$ I want to show that $U \subset S$ open in S $\iff$ there exists an open set $U'$ in $X$ such that $U=U' \cap S$ Here is a little bit of my reasonning: ...
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### Prove that a metrizable space is countably compact iff it is compact.

Prove that a metrizable space is countably compact iff it is compact. ($\Rightarrow$) I let $\{O_i\}$ be a countable open cover for $(X,T)$ with a finite subcover. Let $\{U_i\}$ be an uncountable ...
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### Every 1-Lipschitz function in the closed unit ball has a fixed point

I'm currently trying to solve the following exercise: Let B be the closed unit ball in $\mathbb R^n$ together with the euclidean metric. Show that every 1-Lipschitz function $f:B\to B$ has a ...
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### Completeness of a metric space

I need help solving the following exercise: Show that a metric space $(X, d_X)$ is complete if, and only if, for every isometric embedding $f:X \to Y$ in another metric space $(Y,d_Y)$, it holds ...
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### Geodesic of metric space

The define of geodesic is in below picture which is from Wiki. I don't know why it is generalize of geodesic for Riemannian manifolds. In fact , I can't see it is the shortest when two point are ...
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### $\inf_{x\in A}{\limsup_nd(x_n, x)} = \limsup_n[\inf_{x\in A}d(x_n, x)]$ for compact subset $A$.

let $(X, d)$ be a complete metric space, $A\subset X$ be compact and take a sequence $(x_n) \subset X$\ $A$ as a bounded sequence. Since infimum is independent from n , does the following ...
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### If $\ d(x_n,x)$ exist then $\ (x_n)$ must be converge a point in $X$ ?

Let $\ (X,d)$ be complete metric space, $\ x \in X$ and $\ (x_n) \subset X$ bounded sequence. If the real valued sequence {$\ d(x_n,x)$} convgergent then $\ (x_n)$ must be converge a point in ...
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### An elementary question about real plane metrics

Given the metric $d_p$ on the real plane, i.e., $d_p(x,y)=[|x_1-y_1|^p+|x_2-y_2|^p]^{1/p}$ For which values of $p \geq 1$ is it true that the following set is the usual line segment in the real ...
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### Hausdorff distance on power sets

Consider a general metric space $(S,d)$, with $d$ a $1$-bounded metric, and let $X,Y \subset S$ be two closed subsets of $S$. Notice that $X$ and $Y$ are not compact. Let $\mathcal{P}(X)$ denote the ...
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### When are the distance between points and sets well-defined?

Let $G$ be an open subset of $\mathbb C$. I would like to prove that this set $\{z\in G; d(z,\mathbb C-G)\ge 1/n\}$, where $n\in \mathbb R$, is well-defined. In another words, I would like to know if ...
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### $\overline{X\cap Y}\subset \overline{X}\cap\overline{Y}$ for real numbers, case when $\overline{X\cap Y}\neq \overline{X}\cap\overline{Y}$

My proof for this is similar to this one, but I can't find an example such that $\overline{X\cap Y}\neq \overline{X}\cap\overline{Y}$ for the real numbers.
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### Together with the algebra of cardinal numbers, is there analysis of cardinal numbers? [closed]

Let $C$ be the collection of all cardinal numbers. Is there any norm, inner-product, metric (other than discrete metric), topology(other than discrete, co-finite topology) on $C$, which is very useful?...
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### for $X\subset \mathbb{R}$, $\mathbb{R} = int X\cup Int(\mathbb{R}-X)\cup \partial X$

I need to prove: for $X\subset \mathbb{R}$, $$\mathbb{R} = int X\cup Int(\mathbb{R}-X)\cup \partial X$$ The problem is that all the proofs I've found are for metric spaces, not $\mathbb{R}$ itself, ...