Can open sets be described in-terms of closed sets?

[Alpharup]

Let A be an open set and A=(a,b). Can A be described, as closed set as
"or every x>0, all the elements of closed set [a+x,b-x] are elements of A"?

 

[pwsnafu]

The statement in quotes means
$\forall x>0$, $[a+x, b-x] \subset A$,
which is a true statement.

But
$\forall x>0$, $[a+x, b-x] \subset [a,b]$
is also a true statement, so you can't use it to define A.

 

[Alpharup]

Yes you are right...It just becomes a subset. But is there any way to describe it?

 

[pwsnafu]

Sure. For example, let "A be the union of all $[a+x,b-x]$ with $x>0$." Or "A is the smallest set that contain $[a+x,b-x]$ with $x>0$".

 

[Alpharup]

Yes,,,,the union part...If there were no union, every set (a+x,b-x) is a subset of both (a,b) and [a,b]