Give an example of subsets X and Y of Real number such that all the following conditions are satisfied: (i) X and Y are not open, (ii) X∩Y=∅, (iii) X∪Y is open.

Question

anonymous · Answer

how about X is irrationals in the interval (0,1) and Y are the rationals in (0,1)

anonymous · Answer

thank you...i have to check the conditions first..

anonymous · Answer

sure
it is fairly clear that $X\cap Y=\emptyset$ since one is defined as what the other one isn't

kinggeorge · Answer

Aren't both of those sets open?

anonymous · Answer

not sure what your definition of open is, but it is also true that any open interval around a rational number contains and irrational, and vice versa

anonymous · Answer

@KingGeorge  one definition of open set is that any member of that set is contained in an open interval completely contained in the set. so no, rationals are not an open set

anonymous · Answer

so,rationals number in (0,1) is open set or not??????

anonymous · Answer

no

kinggeorge · Answer

I'm not quite convinced they're closed yet. I could see the rationals being closed, but what about the irrationals?

anonymous · Answer

same thing. any epsilon ball about any irrational number contains infinitely many rationals

kinggeorge · Answer

Look at it this way. We know that a union of a finite number of closed sets is closed. Thus, $X\cup Y$ must be closed if both sets are closed, but $(0, 1)$ is clearly an open set. This is a contradiction.

anonymous · Answer

hold on a sec
i am not suggesting that the rationals are closed

anonymous · Answer

i just said they were not open

anonymous · Answer

problem just says X and Y are not open
it does not say X and Y are closed

kinggeorge · Answer

I see. Yes. You are correct. I was getting my definitions mixed up.

anonymous · Answer

how to prove that the union of X and Y is open?

anonymous · Answer

because early we said that X and Y is not open.....

kinggeorge · Answer

Every number in the set $(0, 1)$ is either rational or irrational. Thus, if we take the union of the two sets of rational and irrational numbers in that interval, we find that we must get every number in the interval $(0, 1)$. Since that set is open, the union is open.

anonymous · Answer

ooo...ok....thanks.....