The notion of $2^S$ in topological space.

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 $$u \subset 2^S$$?

• It means $u\subseteq 2^S$, as it states. You could write that as $u\in 2^{2^S}$ if your prefer. Commented Dec 27, 2018 at 16:42
• Is it right that if $a$ is an element of $u$ , then $a$ is an element of $2^S$ rather than an element of $S$ Commented Dec 27, 2018 at 16:45
• I'm voting to close this question as off-topic because it is not a question about the definition of topological spaces, as stated in the title of the question. Rather, it is a question about the use of notation 2^S. Commented Dec 27, 2018 at 16:54
• Thank you , I've just edited it. Commented Dec 27, 2018 at 16:59

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.
• So , an element in $u$ is a 'set' of $2^S$ rather than an element of $S$ , is it right ? Commented Dec 27, 2018 at 16:48
• I'd rather say that an element of $u$ is a subset of $S$, which is the same thing as asserting that it is an element of $2^S$. Commented Dec 27, 2018 at 16:53