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

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1. Apr 9, 2016

### 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"?

2. Apr 9, 2016

### 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.

3. Apr 9, 2016

### Alpharup

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

4. Apr 9, 2016

### 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 $x>0$".

5. Apr 9, 2016

### 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]