# 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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### A metric between functions on $\mathbb{R}^2$

I want to measure the distance between functions $f$ and $g$ (not necessarily continuous) on a bounded subset $M\subset\mathbb{R}^2$. I assume $f$ and $g$ are locally integrable and bounded on $M$. ...
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### $A,B$ closed subsets of $\mathbb R^n$ , when can we say (other than compact-ness of $A$ or $B$ ) $\exists b \in B$ such that $dist(A,B)=dist(b,A)$ ?

Let $A,B$ be disjoint closed subsets of $\mathbb R^n$ , when can we say ( weaker than compact-ness of $A$ or $B$ ) that there exist $b \in B$ such that $dist(A,B)=dist(b,A)$ ? I know that if $A,B$ are ...
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### If $Z(f)$ is the zero set, prove that $Z(f)$ is closed

Introduction: Exercise from Principles of Mathematical Analysis, third edition (Rudin), page 98. Exercise: Let $f$ be a continous real function on a metric space $X$. Let $Z(f)$ (the zero set of ...
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### Is the empty set an open ball in a metric space?

Problem Let $(X,d)$ be a metric space where $X$ is a non-empty set. Is the empty set an open ball in $X$? I think that it is true because if $X=\mathbb{R}$ with the usual metric then for all ...
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### proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$

Intuitively this proposition seems true, but I've been told that is not a trivial thing to prove. Is there any simple proof (or counter-example) for the proposition: Consider the closed ball of ...
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### EDITTED: Find all values of $a$ and $b$ so that $ax^n+b\cos\left(\frac{x}{n}\right)$ is Cauchy.

For each $n\in\mathbb{N}$ let $$f_n(x)=ax^n+b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1].$$ Find all values of $a$ and $b$ for which $(f_n)$ is a Cauchy sequence in $C[0,1]$, the space of ...
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### Any bi-invariant distance on a group is inverse-invariant?

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...