Compactness and FIP related problem

[librastar]

Homework Statement

This question is related to Topology.

Let X be a compact space and let {C_a|a$$\in$$A} be a collection of closed sets, closed with respect to finite intersections. Let C = $$\cap$$C_a and suppose that C$$\subset$$U with U open. Show that C_a$$\subset$$U for some a.

The Attempt at a Solution

Here is how my solution goes:

Consider the complement of C_a, X - C_a, is open.
Since X is compact, $$\cup$$(X-C_a) is the open cover of X and $$\cup$$(X-C_i) for i = 1,2,...,n is a finite subcover of X such that X = $$\cup$$(X-C_i).
Now since X is compact and by the finite intersection property, C is nonempty.

But here is where I got stuck...I don't know how to continue to finish the problem.
I think this may be caused by mistakes in my reasoning, but I can't spot it.
Please help me on this question, any help is welcomed.

Thanks in advance.

 

[Office_Shredder]

The closed under finite intersections is important for more than just the fact that C is non-empty. For example, let $$X=[-2,2]$$, $$C_1 = [-2,1]$$ and $$C_2=[-1,2]$$. Then $$C=[-1,1]\subset (-\frac{3}{2}, \frac{3}{2} )=U$$ but neither $$C_1$$ or $$C_2$$ are contained in $$U$$

 

[librastar]

Sorry but I still don't quite get it...

So the compactness of X comes into play because now I can generate a open set U that is "big" enough to contain some or all of C_a?