Explanation of 'Infinite collection of intervals'?

Here's a situation which is perhaps more familiar to you. Sometimes, in calculus, you run across a question which starts like this:

Given an infinite sequence $x_n$, $n=1,2,...$ of real numbers,...

When you see this, you know that $x_1$ can be an arbitrary real number, and $x_2$ can be an arbitrary real number, and the same for $x_3$, $x_4$, and for every other number in this sequence. This kind of question will be telling you some information about an arbitrary sequence of real numbers. For example, this is how you would start the definition of the limit of the sequence. Or, this is how you would start the theorem that the limit of a sum of two sequences (this sequence $x_n$ and another sequence $y_n$) is equal to the sum of the limits. Or something else, who knows.

So, suppose you run across a question which starts like this:

Given an infinite collection $A_n$, $n=1,2,...$ of intervals in the real line...

When you see this, you know that $A_1$ can be an arbitrary interval in the real line, and $A_2$ can be an arbitrary interval in the real line, and the same for $A_3$, $A_4$, and for every other interval in this collection. This kind of question will be telling you something about an arbitrary collection of intervals in the real line. So, for example, your question is giving you the definition of the intersection of an arbitrary collection of intervals in the real line.

Each $A_n$ is some interval. For instance, you may have $A_1=[0,2)$, $A_2=\left(-3,\frac12\right)$, and so on.

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).