Definition: A topological space is a set $S$ together with a collection $u$ of subsets of $S$ (that is, $u$ is a subset of $2^S$ ) satisfying the following conditions: ...
This is a definition of topological space in "Lecture Notes on Elementary Topology and Geometry" on page 6. I'm quite confused about the words in the bracket. Since here, $2^S$ denotes the collection of all the subsets of $S$, how to understand the subset of the collection of all subset of $S$? Does it mean that $u \in 2^S$ rather than $u \subset 2^S$?
No. It means that $u\subset2^S$. In other words, $u$ consists of subsets of $S$. Asserting that $u\in2^S$ would mean that $u$ is a subset of $S$ instead.