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Let $F_k\subset \Bbb{R}$ be an open interval in $\Bbb{R}$, and $x\in \Bbb{R}$ a point. How is $d(x,F_k)$ defined? I came across this notation in my textbook and it is confusing me. Is $d(x,F_k)=\min\{d(x,y)\},\forall y\in F_k$? And if this definition is the correct one, then is it valid only for bounded intervals $F_k\subset \Bbb{R}$?

Thank you!

hudialala
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    The distance between the point and the set is defined as $d(x,F_k)=\inf{d(x,y)\mid y\in F_k}$. But $F_k$ can be replaced by any set you want. – Stefan Hamcke Aug 11 '13 at 13:04
  • $F_k$ is not necessarily bounded. For example, $F=]4;+\infty[$ and $d(2,F)=2$. – user5402 Aug 11 '13 at 13:14

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As Stefan H. noted, $d(x,F)=\inf\{|x-y|:y\in F\}$. This infimum is finite for any nonempty set, bounded or not. It is guaranteed to be attained by some $y\in F$ if $F$ is closed (this works because we are in the Euclidean space, where bounded+closed=compact).

When $F$ is an open interval $(a,b)$ on $\mathbb R$, one can find a more explicit formula for $d(x,F)$: $$d(x,F)=\begin{cases} a-x,\quad & x<a \\ 0 ,\quad & a\le x\le b \\ x-b,\quad &x>b \end{cases}$$