Open set in a topological space

[ehrenfest]

Homework Statement

If U is open set in a topological space such that U = A union B, where A and B are disjoint, do both A and B have to both be open?

I think that they do not, but I cannot think of a counterexample...perhaps (-1,0) union [0,-1).

OK. That's a counterexample. So, now the question can both A and B not be open?

Homework Equations

The Attempt at a Solution

 

[Office_Shredder]

What about in R², trying the same trick only with say, two rectangles, one on (0,1/2)x(0,1) and one on (1/2,1)x(0,1), and then give each half of the boundary between them

 

[Dick]

Or take U to be (0,1), A=rationals in U and B=U-A. Neither is open.

 

[HallsofIvy]

No need to get as complicated as "rationals" and "irrationals". Take A= [0, 1], B= [2, 3] so that both are closed. Then the intersection is the empty set which is open!

Or, since the empty set is also closed, you might prefer A= [0, 3/2), B= (1, 2]. Neither is open but their intersection is (1, 3/2) which is open.

 

[ehrenfest]

The union of the two sets is open, not the intersection.