# Math Help - Is it an open set?

1. ## Is it an open set?

If O is open and F is closed. Is O/F open?

2. Originally Posted by Cerny
If O is open and F is closed. Is O/F open?

What do you mean by O/F??

Tonio

3. x is in O/F iff x is in O but not in F.

I understand this should be a backslash. I put the blame on my keyboard...

4. If there is no such x, then O/F is an empty set. Hence O/F is open. I couldn't show that the result holds when O/F is nonempty.

5. Hello,

By definition of the set difference, $O\setminus F=O\cap F^c$
But by definition of a closed set in a topology, $F^c$ is an open set.
And from one of the axioms of a topology, the intersection of two open sets is still an open set.