I was working with the Cantor Set today and its construction prompted me to think about infinite unions/intersections a little more than usual.
What exactly does an infinite union/intersection represent?
I think of $$\bigcup\limits_{i=1}^{\infty} F_{i}$$ as $$\lim \limits_{n \to \infty}\bigcup\limits_{i=1}^{n} F_{i}$$
I understand what is going on at each finite stage, but I'm confused about what is occurring at the "end" of the process.
For example, we define the Cantor set to be the infinite intersection of increasing middle thirds sets, but is the Cantor set itself the "limit set" of this process? And if so what does it mean for a set to be a limit?
I appreciate any insight. Thanks.
Best Answer
$\bigcup_{i=1}^\infty F_i$ is the set defined by the property $$ x\in \bigcup_{i=1}^\infty F_i \iff\mbox{$x\in F_i$ for some $i\in\{1,2,\ldots\}$}.$$
The thing that separates the infinite situation from the finite one is that we need to consider a countably infinite collection of sets $\{F_1,F_2,\ldots \}$ instead of only a finite collection $\{F_1,F_2,\ldots, F_n\}.$
So we don't really need to think of a limiting process here... it's more about the "completed" infinity of the collection of $F_i.$ Consider that in theory the $F_i$ might bear no obvious relation to one another, though in practice of course they often do. However, if it's useful, we can imagine building up the union progressively, by adding first the elements of $F_1,$ then all the elements of $F_2$ that weren't already included, then all the elements of $F_3,$ and so on.