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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 $u \subset 2^S$?

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    $\begingroup$ It means $u\subseteq 2^S$, as it states. You could write that as $u\in 2^{2^S}$ if your prefer. $\endgroup$ Commented Dec 27, 2018 at 16:42
  • $\begingroup$ Is it right that if $a$ is an element of $u$ , then $a$ is an element of $2^S$ rather than an element of $S$ $\endgroup$
    – J.Guo
    Commented Dec 27, 2018 at 16:45
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    $\begingroup$ 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. $\endgroup$
    – user126154
    Commented Dec 27, 2018 at 16:54
  • $\begingroup$ Thank you , I've just edited it. $\endgroup$
    – J.Guo
    Commented Dec 27, 2018 at 16:59

1 Answer 1

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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.

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  • $\begingroup$ So , an element in $u$ is a 'set' of $2^S$ rather than an element of $S$ , is it right ? $\endgroup$
    – J.Guo
    Commented Dec 27, 2018 at 16:48
  • $\begingroup$ 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 $2^S$. $\endgroup$ Commented Dec 27, 2018 at 16:53

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