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1.Let $X$ be a set.

Let $\mathscr{A}$ be a family of subsets of $X$.

Here, what does 'a family' means precisely?

Does this mean $\{f(\alpha)\}_{\alpha\in A}$ for some $A\subset P(X)$ and $f:A\rightarrow P(X)$?

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A "family" of sets is just a set of sets. People use the word "family" or "collection" to avoid becoming confused or confusing others. So yes, $\mathcal{A}$ is a subset of the power set of $X$. – Stefan Smith Oct 30 '13 at 0:57
Basically it means $\mathscr{A} \subset {\cal P}(X)$. – copper.hat Oct 30 '13 at 0:58
There is no logic in my family. – copper.hat Oct 30 '13 at 0:59
Because one doesn't want the understanding of a topology to depend on silly stuff about empty intersections. – André Nicolas Oct 30 '13 at 1:12
You're quite right that clause 1 in the definition of "topology" is redundant. It's included for the benefit of people who don't like to think about the empty family or who may be confused about what its union and (especially) its intersection are. – Andreas Blass Oct 30 '13 at 1:13
up vote 2 down vote accepted

Yes; the empty set is indeed a family of sets.

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