# Open sets

1. Feb 13, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. The attempt at a solution

Last edited: Feb 13, 2008
2. Feb 13, 2008

### 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.

3. Feb 13, 2008

### ehrenfest

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

4. Feb 13, 2008

### Mathdope

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

5. Feb 13, 2008

### ehrenfest

Sorry. Reread the EDIT. I am foolish.

6. Feb 13, 2008

### NateTG

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.