TITLE: Is it weird to say $A \in B \in C$?
QUESTION [5 upvotes]: I've just noticed that I've never seen any text say $A \in B \in C$, which is why when writing it myself it immediately looked weird.
For context, I was proving a result about the topology $\mathcal{T}$ generated by a basis $\mathcal{B}$. In particular, the fact that $\displaystyle\bigcup_{i \in \mathcal{A}} U_i \in \mathcal{T}$. I came to a line that said:

Then $\exists i$ such that $x \in U_i$ and $U_i \in \mathcal{T}$.

In my own proof I had written

$x \in U_i \in \mathcal{T}$.

Why has the proof been split into $x \in U_i$ and $U_i \in \mathcal{T}$? Is there any purpose for this?

REPLY [2 votes]: It is not any more weird than writing $a<b<c$, although strictly speaking we should not do that either.   However, since its generally understood to mean $a<b$ and $b<c$ it is a convenient abbreviation.
We do often write $a\in B\subset C$ with the understanding that this means $a\in B$ and $B\subset C$, and therefore $a\in C$ also.
However, the statement $a\in B\in \mathcal C$ "looks weird" because "$\in$" is not transitive.   It would seem to imply that $a\in \mathcal C$, which is not necessarily the case.   Thus it is more conventional not to abuse the notation and instead write it in full: $a\in B$ and $B\in \mathcal C$