1. ## collection of subsets

Let B be the collection of subsets of R of the form [a,b) for a<b, nulset, and the whole set R.

Show that if A and C are elements of B then A intersect B is in B.
and
Show that if {A_i| 0 <= i <= n} is a finite collection of subsets of B then the collection of intersections (from i=1 to n) of A_i is in B.

2. Just verify the different possible cases

If $A$ or $C$ is $\emptyset$, then their intersection is the $\emptyset$ and is in B.

If $A$ or $C$ is $\mathbb{R}$, then their intersection is the other set and is in $B$.

If $A=[a,b[$ and $C=[c,d[$, then $A\cap C=[\max (a,c),\min (b,d)[$ if $b, and $A\cap C=\emptyset$ else.

An induction is enough to generalize that result for your second question.

3. Originally Posted by EricaMae
Let B be the collection of subsets of R of the form [a,b) for a<b, nulset, and the whole set R. Show that if A and C are elements of B then A intersect B is in B.
You must mean C where you have B above.
$A=[a,b)\;\&\; C=[c,d)$. Then let $\alpha=\max\{a,c\}\;\&\; \gamma=\min\{b,d\}$.
If $\gamma \leq \alpha\text{ then } A\cap C = \emptyset \in B$.
Else $A \cap C = [\alpha , \gamma) \in B$.
You do the general case.