Can $\infty$ be used to index a set? For example $C_{\infty}$ or something like $\bigcap_{i=0}^{\infty}C_i=C_0\cap C_1\cap\cdots\cap C_{\infty}$

Question

Does
$$\bigcap_{i=0}^{\infty}C_i=C_0\cap C_1\cap\cdots\cap C_{\infty}?$$

I'm pretty sure that it is correct, but what does $C_{\infty}$ actually mean, I haven't studied infinity yet so I don't know if it's valid or what it means.

Answer 1

There are two questions here:

1. Can $\infty$ be used as an index?

   Sure. Any symbol at all can be used as an index. It is just notation. For example, it is very common to use $\infty$ as the index of a limiting object. In the question, it would not be unreasonable to write $$ C_{\infty} = \bigcap_{j=0}^{\infty} C_j = C_0 \cap C_1 \cap C_2 \cap C_3 \cap \dotsb. $$

2. Does the notation $C_0\cap C_1 \cap C_2 \cap \dotsb \cap C_{\infty}$ make sense?

   No. This is not really meaningful notation (except, possibly, in some very esoteric contexts which, if you are asking this question, are not relevant). By convention, the notation $$\bigcap_{j=0}^{\infty} C_j$$ means "take the intersection of all of the sets $C_j$, where $j$ is any natural number (or zero). Another way of writing this is $$ \bigcap_{j\in\mathbb{N}} C_j. $$ There is no set labeled $C_\infty$ in this collection of sets which are being intersected.

Answer 2

Infinity can be used to index a set, but it's not very common.

The notation you are asking about means $$ \bigcap_{i=0}^{\infty} C_i =C_0\cap C_1\cap \cdots $$ i.e. the set of points that are found in every $C_i.$