If O is open and F is closed. Is O/F open?
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Originally Posted by Cerny If O is open and F is closed. Is O/F open?
What do you mean by O/F??
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...
Last edited by Cerny; September 18th 2010 at 10:32 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.
Last edited by Cerny; September 18th 2010 at 08:48 PM.
By definition of the set difference,
But by definition of a closed set in a topology, 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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