The difference between a family of sets and an indexed family of sets

[Simonel]

What is the difference between family of sets and indexed family of sets ??

 

[fresh_42]

What is the difference between family of sets and indexed family of sets ??

None, one is labeled (indexed) and the other one is not, which makes it difficult to refer to a subset of them.

Sometimes the index is rather formal without any specification.

E.g. it's easy to index the family $\{\,[-\frac{1}{n},\frac{1}{n}] \,:\,n \in \mathbb{N}\,\}$ by $n \in \mathbb{N}$ such that we get $\{\, [-\frac{1}{n},\frac{1}{n}]\,\,: \,n \in \mathbb{N}\}=\{\,I_n\,\}_{n \in \mathbb{N}}$ with $I_n :=[-\frac{1}{n},\frac{1}{n}]$. Occasionally such families are written as a union.

On the other hand, if we define the family of all subsets of $\mathbb{R}$ which contain $\pi$, then there is no explicit indexing possible. Yet we can write $\mathcal{F} = \{\, S \subseteq \mathbb{R}\, : \, \pi \in S \,\}$ as $\mathcal{F} = \{S_\iota\}_{\iota \in I}\; , \;\pi \in S_\iota$ which gives us a formal index set, although we cannot name it.