If a collection on $\Omega$ is closed under finite intersections, then can we conclude that it contains $\Omega$? Here an example that illustrates my question.

A $\pi$-system $\mathcal P$ on set $\Omega$ is defined here to be a collection that is not empty and satisfies the rule $A,B\in\mathcal P\implies A\cap B\in\mathcal P$.

Then they go on by saying that the collection is closed under finite intersections. 

But wait a minute... Isn't $\varnothing$ a finite set? 

Interpreting the empty intersection as $\Omega$, this can make people think that $\Omega\in\mathcal P$, which is not a consequence of the definition.

($\varnothing\subseteq\mathcal P$ and it is vacuously true that $\omega\in P$ for every $\omega\in\Omega$ and every $P\in\varnothing$)

Personally I would rather go for "closed under binary intersection" keeping in mind of course that this implies "closed under non-empty finite intersections".

Thanks in advance for sharing your thoughts on this.
I agree that for maximal naturality and elegance we should demand that $\Omega$ is in every $\pi$-system.

However, the notion of a $\pi$-system is already not a particularly natural one and is only really used in the $\pi-\lambda$ theorem that says that any $\lambda$-system contains a $\pi$-system then it also contains the $\sigma$-algebra generated by that $\pi$-system. Since $\sigma$-algebras and $\lambda$-systems already have to contain $\Omega$ this theorem would only be made weaker by demanding that $\Omega$ be in the $\pi$-system.

As it is we have made the $\pi-\lambda$ theorem slightly stronger, and at not much expense of elegance, since the notion of a $\pi$-system would is pretty ugly anyway.
They don't mean intersections of finite sets, they mean intersections of finite numbers of sets. For instance:
if A,B, and C are sets in P, then $A\cap B\cap C\in P$
In general, this is true because, the base case follows from the definition of P and if $\cap$ {$A_1,A_2,....A_k$} $\in P$, for $k\in \mathbb N$, and $A_{k+1}$ is any set, ($\cap$ {$A_1,A_2,....A_k$})$\cap A_{k+1}$ is just an intersection of 2 sets, which we know is in P by the definition of P. Thus, by induction on the number of sets in an intersection P is closed under the intersection of any finite number n of sets: i.e. closed under finite intersection.

Also, from the information you provided, it doesn't even follow that the empty set has to belong to a given pi system of a set omega, although there are surely some pi systems that do; for instance, the pi system {O}. Clearly this is a pi system of any set omega since it's a non-empty collection of subsets of omega that is closed under intersections of any finite number of sets within it.