# An open set whose complement isn't open

#1
Consider Z with the evenly spaced integer topology, I want to find a set whose complement isn't open.

Here's a sketch of how I think one can find such a set

Consider {0}, by definition of the topology this isn't open, I show that it's complement is open:

Z/{0} is just the union over (a+bZ) with b not dividing a=/=0, b=/= 0, all of which are open therefore Z/{0} is open and the set I am looking for.
0
3 years ago
#2
(Original post by Ryanzmw)
Consider Z with the evenly spaced integer topology, I want to find a set whose complement isn't open.

Here's a sketch of how I think one can find such a set

Consider {0}, by definition of the topology this isn't open, I show that it's complement is open:

Z/{0} is just the union over (a+bZ) with b not dividing a=/=0, b=/= 0, all of which are open therefore Z/{0} is open and the set I am looking for.

Looks OK to me; you could be more specific about a union that makes up Z\{0}...
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