TITLE: Explanation of 'Infinite collection of intervals'?
QUESTION [5 upvotes]: While doing a course, a questions tells me the following:

Given an infinite collection $A_n$, $n=1,2,\dots$ of intervals of the real line, their intersection is defined to be $\bigcap^∞_{n=1}A_n=\{x\mid(\forall n)(x\in A_n)\}$.

What does this mean? 
Is $A_n$ the collection of $(1,2),(2,3),(3,4),\dots$?
Or is it $[1,2],[2,3],[3,4],\dots$?
Or is it something else entirely?
This may be a very silly question, but I was taught real analysis in a lecture half an hour long, so my knowledge is extremely lacking.
Please help.

REPLY [2 votes]: Imagine the following situation, at first:

Given a finite collection of intervals, $A_1,A_2,\dots,A_k$, we define the intersection of these intervals to be:
$$\bigcap_{i=1}^kA_i:=\{x\in\mathbb{R}|x\in A_i\text{ for each }i=1,2,\dots,k\}.$$

So, you are given some intervals, whatever they may be. For instance, you may be given:
$$\begin{align*}
A_1&=[0,3)\\
A_2&=(-1,5)\\
A_3&=(0,7)\\
A_4&=[1,5]\\
\end{align*}$$
So, their intersection is:
$$\bigcap_{i=1}^4A_i=(0,3),$$
which happens to be an open interval. You may be given:
$$\begin{align*}
A_1&=[0,3)\\
A_2&=(-1,1)\\
A_4&=[1,5]\\
\end{align*}$$
So, their intersection is:
$$\bigcap_{i=1}^4A_i=\varnothing,$$
which is the empty set.
Now, in general, the notion of a collection means that we choose some objects of some kind, or with some specific property(ies). In our case we have chosen a finite collection of some objects named intervals, so, we have in our hands some (5 or 6 or 2,054; finite anyway) sets of numbers, some parts of the real line.
To tackle your question, exactly, when talking about infinite families, the only thing that changes is the "quantity" we have in our hands. An infinite family contains infinitely many objects of that certain kind. For instance, in the case of the infinite intersection, we have that:

Given an infinite collection of intervals, $A_1,A_2,\dots$, we define the intersection of these intervals to be:
$$\bigcap_{i=1}^\infty A_i:=\{x\in\mathbb{R}|x\in A_i\text{ for each }i=1,2,\dots\}.$$

So, you may be given the following infinite collection of intervals:
$$\begin{align*}
A_1&=\left[0,1\right)\\
A_2&=\left[0,\frac{1}{2}\right)\\
A_3&=\left[0,\frac{1}{3}\right)\\
\vdots&=\vdots\\
A_k&=\left[0,\frac{1}{k}\right)\\
\vdots&=\vdots
\end{align*}$$
In this case you can describe this collection of intervals in a nice way, since there is some kind of "normality". Also, their intersection is:
$$\bigcup_{i=1}^\infty A_i=\{0\}.$$
But, a collection of intervals can be very "random". For instance. pick two real numbers - whichever you want - let $a_1,b_1$ with $a_1<b_1$ and let $A_1=(a_1,b_1)$. Then, again, pick two real numbers randomly, let $a_2,b_2$ and let $A_2=(a_2,b_2)$. Then again, then again, then again, then again,... infinitely many times (you can choose infinitely many numbers; it is like choosing two sequences $a_n,b_n$ with $a_n<b_n$ for each $n\in\mathbb{N}$).
You may also meet something like the following: Let $A_1,A_2,\dots$ be an infinite collection of all the intervals with their limits being rational numbers.
As you see, we can create an infinite collection of such examples... (humour intended here).