# 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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### Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ This is the $2$-norm on the $1$ ...
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### Showing two norms is not equivalent

Define the norms as $||f||_u=sup_{x\in[a,b]}\{|f(x)|\}\ \ \ \ \ \ \ ||f||=||f||_u+||f'||_u$ Show that $||\cdot||$ and $||\cdot||_u$ is not equivalent I've found a sequence for which $||\cdot||_u$ ...
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### Metric on the profinite completion of the integers?

The p-adic integers come with a metric and associated topology, both of which can be restricted down to the integers. Does this also apply to the profinite completion of the integers? Do they have ...
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### Metric equivalent same topology

Let $d_1=|x-y|$ and $d_2=|φ(x)-φ(y)|$ where $φ(x)=\frac{x}{(1+|x|)}$ I must proof $d_1$ and $d_2$ define the same topology over $R^2$ I want some hint. just some indication or méthodes .
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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, ...
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### Equivalence of precompactness and completely boundedness.

Definitions: The set $A\subset X$ is called completely bounded if $\forall \epsilon >0 \ \exists x_1,...,x_k \in A$ s.t. $A \subset \bigcup_{i=1}^k B(x_i,\epsilon)$. The set $A$ is called ...
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### Question regarding metric spaces and union of balls

If $X$ is a compact space and $\epsilon > 0$, I want to show that there exists $n$ point $x_1, x_2, ... x_n$ such that $\bigcup_{i=1}^n B(x_i, \epsilon) = X$ I am not sure how to do that. ...
If $X$ is a metric space and $x_0\in X$. Let $x$ and $x'$ be any points of $X$. I want to unerstand why the following inequality is correct: $d(x,x_0)-d(x',x_0) \leq d(x,x')$ I understand that if we ...
### If $S \subset X$ does a subsequence of $S$ converge in $S$ or in $X$?
Let $X$ be a metric space and let $S\subset X$ be a compact space. By definition, $S$ is compact implies that for all sequences $(x_n)$ of $S$, there exists a subsequence $(x_{n_{\alpha}})$ that ...