Are Disjoint Sets in a Topological Space Always Open When Their Union is Open?

[ehrenfest]

Homework Statement

If you have a collection of disjoint open sets in a general topological space whose union is open, is it true that each of them individually must be open? Why?

EDIT: this makes absolutely no sense. here is what I meant to ask:
EDIT:If you have a collection of disjoint sets in a general topological space whose union is open, is it true that
EDIT:each of them individually must be open? Why?

Homework Equations

The Attempt at a Solution

 

[d_leet]

As stated the answer is trivially yes because they are disjoint and OPEN. Did you really mean to ask if you had a collection of disjoint set's whose union in a topological space is open is each necessarily open. In that case the answer is false consider the trivial topology on {a,b,c}. Then the collection {{a}, {b}, {c}} is a collection of disjoint sets whose union is open but none of the ndividual sets is open.

 

[ehrenfest]

As stated the answer is trivially yes because they are disjoint and OPEN. Did you really mean to ask if you had a collection of disjoint set's whose union in a topological space is open is each necessarily open. In that case the answer is false consider the trivial topology on {a,b,c}. Then the collection {{a}, {b}, {c}} is a collection of disjoint sets whose union is open but none of the ndividual sets is open.

What if we are in a locally path connected and path connected space?

 

[Mathdope]

What are the original sets that comprise the union? They can't be open individually if you are being asked that very question.

 

[ehrenfest]

What are the original sets that comprise the union? They can't be open individually if you are being asked that very question.

Sorry. Reread the EDIT. I am foolish.

 

[NateTG]

Sorry. Reread the EDIT. I am foolish.

Consider any non-open set X and it's complement \bar{X}.
Their union is clearly open since it's the entire space, they're disjoint by construction, and X is non-open by hypothesis.