Thread: open set

Are the following are open sets in . In each case say why?

1.[0,)

2. -


3.

For 1. I know its not an open set because the complement is open so [0,) is closed.


But need help for 2 and 3.

Thanks

Originally Posted by 1234567

Are the following are open sets in . In each case say why?

1.[0,)

2. -


3.

For 1. I know its not an open set because the complement is open so [0,) is closed.


But need help for 2 and 3.

Thanks

How are you defining open here? the natural way? open subsets of the reals are open intervals? if so,

yes, 1. is closed, because its compliment is open.

For 2, notice that you have the set:



For 3, is a set of points on the real line... but be careful, is dense in , so you're going to want to come from the perspective of boundary points, etc

Yes open subsets of the reals are open intervals.

Thanks for the help.

For 3. I know the complement is the set of all irrational numbers which is not open so do that mean is also not open.

Originally Posted by 1234567

Yes open subsets of the reals are open intervals.

Actually any open set is is the union of open intervals.