Definition of $\pi$ and d -systems. Definition: Let $\Omega$ be a sample space.
a) A d-system is a family of subsets containing $\Omega$ and closed under proper difference 
(if A,B $\in\mathcal D$ and A $\subseteq$ B, then B \ A $\in\mathcal D$ ) and countable increasing union.
b) A $\pi$-system is a family of subsets closed under finite intersection. 
What does it mean to say something is closed under finite intersection? And when something is a countable increasing union? 
Could someone please provide a simple example of a $\pi$-system and a d-system? 
 A: What does it mean to say something is closed under finite intersection? This means that if $A,B$ are elements of a $\pi$-system $\mathcal{P}$, then $A \cap B$ is in $\mathcal{P}$ as well (and by induction, if $A_1$, ... $A_n \in \mathcal{P}$ then $\bigcap_{i=1}^n A_i \in \mathcal{P}$.
To be closed under countable increasing union means that if $A_1$, $A_2$, ... is an increasing sequence of elements of $\mathcal{D}$, then $\bigcup_{n \geq 1} A_n \in \mathcal{D}$. Here to be increasing means that $A_1 \subset A_2 \subset A_3 \subset \cdots$.
To give some examples:


*

*The set of all subsets of $\Omega$ is both a $d$-system and a $\pi$-system;

*More generally, any $\sigma$-algebra on $\Omega$ is both a $d$-system and a $\pi$-system;

*The set of all intervals $[a,b)$ with $a,b \in \mathbb{R}$ is a $\pi$-system because $[a_1,a_2) \cap [b_1,b_2) = [\max(a_1,b_1),\min(a_2,b_2))$, but is not a $d$-system because for instance $[0,3) \backslash [1,2)$ is not an interval;

*If $\Omega$ has finite even size, then the collection of all subsets of $\Omega$ having an even number of elements is a $d$-system (because the complement or union of two sets with even size again has even size), but not a $\pi$-system (because, for instance, $\{a,b\} \cap \{a,c\} = \{a\}$).

A: A topology is a $\pi$ system.  Notice that if $\{A_n\}$ is a finite family of open sets, the intersection is open as well.  This means that a topology is closed under "finite intersections."  Compare this to being closed under "arbitrary unions" and "arbitrary intersections".  We can easily come up with a family of open sets whose intersection is not open.  The family would necessarily be infinite, though.  But no matter how big the family is (countable or even uncountable), the union of open sets is open.
The condition on $d$-systems means that if you have a sequence of increasing sets (so that $A_i \subset A_j$ if $i < j$) then the union of all the $A_i$ is in the $d$-system. If you wanted to put this into words, you might say a $d$ system is closed under countable increasing unions -- i.e., you would recast the condition as saying that a $d$-system is closed under a function $f\colon \{A_i\}_{\subset} \rightarrow \Omega$ defined by $\bigcup_i\ A_i$.  Hopefully you understand the notation -- it is meant to be suggestive, not a representation of any "real" usage.  The point is, of course, that it's closed under unions on a restricted domain -- just the increasing sequences.
Again, a topology is a $d$-system.  In particular, since the union of any sequence of open sets is open, surely the union of an increasing sequence of open sets is open.
