**Theorem**.

*If {Kα} is a collection of compact subsets of a metric space X such that the intersection of every finite subcollection of {Kα} is nonempty, then ∩Kα is nonempty.* **Proof**. Fix a member

*K*1 of

{*K**α*} and put

*G**α*=*K**c**α* [this denotes the complement of

*K**α*]. Assume that no point of

*K*1 belongs to every

*K**α*. Then the sets

*G**α* form an open cover of

*K*1 [this took me a bit but that's by the last assumption]; and since

*K*1 is compact, there are finitely many indices

*α*1,…,*α**n* such that

*K*1⊂⋃*n**i*=1*G**α**i* [so far so good]. But this means that

*K*1∩*K**α*1∩…*K**α**n*

is empty, in contradiction to our hypothesis.

The area enclosed by the green is , by the fuschia and together they make up the set The blue lines indicate the area of . And is the union of the blue shaded area and the part of $K_2$ not occupied by $K_1$ (which I didn't draw out as it looked too messy for one picture)

Line for line of the proof;

Fix a member of $K_1$ of $\{K_\alpha\}$...

I think I'm good on this line. It just means call the green blob's insides $K_1$

Assume that no point of $K_1$ belongs to every $K_\alpha$.

This means that $K_1 \cap K_2$ is empty. So the bit of the green that overlaps the fuschia is empty.

Then the sets $G_\alpha$ form an open cover of $K_1$;

As blue area and the compliment to $K_1$ definitively covers everything but $K_1 \cap K_2$, but that's empty, so this looks acceptable to me.

and since $K_1$ is compact, there are finitely many indices

I think this means that there is a collection of open sets which cover $K_1$. This is only true because we've removed that chunk with the; "Assume that no point of $K_1$ belongs to every $K_\alpha$.".

But this means that $K_1 \cap K_2 \cap ... \cap K_{\alpha_n}$ is empty, in contradiction to our hypothesis.

I don't see how this contradicts anything. He assumed the intersection of all K's was empty and then showed that the intersection of all K's is empty. It looks like he's written $A \land A$ conclude $\perp$ which is nonsense.

Where are my errors?