Is it an open set?

**Cerny** · September 18th 2010, 06:55 PM

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

**tonio** · September 18th 2010, 07:30 PM

Originally Posted by Cerny

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

What do you mean by O/F??

Tonio

**Cerny** · September 18th 2010, 07:34 PM

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...

**Cerny** · September 18th 2010, 07:36 PM

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.

**Moo** · September 19th 2010, 02:29 AM

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.

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