# Fine Families Proof

1. Sep 27, 2009

### cmajor47

1. The problem statement, all variables and given/known data
Recall that when F is fine and consistent, we define F* to be the family of all intervals which intersect every member of F, that is F*={K where K intersects every I in F}.
Prove that F* is itself fine and consistent.

2. Relevant equations
If F is a consistent family of intervals, so is the family of all finite intersections of members of F, and it contains F as a subfamily.
If F is a fine and consistent family, define F* to be the family of all [a, b] that intersect every member of F (or, equivalently, all [a, b] such that a$$\leq$$F$$\leq$$b).

3. The attempt at a solution
Proof: Let F be fine and consistent, and let F* be the family of all intervals which intersect every member of F, that is F*={K where K intersects every I in F}.
Since F is fine and consistent and F* contains F as a subfamily, F* is itself consistent.
Also, since F* contains every interval of F, F* itself will be fine since F is assumed to be.
Therefore, F* is itself fine and consistent.

I just wanted to make sure that this proof wasn't missing any logical steps.