# Question about open extensions of sets in metric spaces.

• Jun 14th 2013, 05:28 AM
Question about open extensions of sets in metric spaces.
Let $(X,d)$ be an arbitrary metric space. For every $A\subset X$ and $t>0$, we define the open $t$-extension of $A$ as the set

$A_t=\{x\in X:d(x,A),

where $d(x,A)=\inf\{d(x,y):y\in A\}$.

It seems intuitive that, given $A\subset X$ and $s,t>0$, we should have that $A_{t+s}=(A_t)_s=(A_s)_t$. However, I haven't been able to even begin to construct a rigorous argument to prove it. Any hint or proposed direction would be greatly appreciated (or counterexample in the event that it is false).
• Jun 14th 2013, 06:00 AM
Plato
Re: Question about open extensions of sets in metric spaces.
Quote:

Let $(X,d)$ be an arbitrary metric space. For every $A\subset X$ and $t>0$, we define the open $t$-extension of $A$ as the set
$A_t=\{x\in X:d(x,A),
where $d(x,A)=\inf\{d(x,y):y\in A\}$.
It seems intuitive that, given $A\subset X$ and $s,t>0$, we should have that $A_{t+s}=(A_t)_s=(A_s)_t$. However, I haven't been able to even begin to construct a rigorous argument to prove it. Any hint or proposed direction would be greatly appreciated (or counterexample in the event that it is false).

When you say that you are unable "to construct a rigorous argument", does that mean that you cannot even start?
Can you show that $(A_t)_s\subseteq A_{t+s}~?$

Before I work on this, please post what you have been able to do.