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I'm strugling with countability over indexed sets compared to ordinary sets. Basically we say that any set $A$ is countable if there is a bijective function $f$ such that $f:\mathbb{N}\longrightarrow A$. Here my problem is that I don't know if there is a similar standard definition when we talk about families of the form$\{a_i\mid i\in I\}$. Essentially what I'm looking for is something like:

"A family of sets is countable if and only if..."

Intuitively I'd say that a family of sets $\{a_i\mid i\in I\}$ is countable if there is bijection between $I$ and $\mathbb{N}$, but here I don't know if this is a consequence of the definition or this can be admited as such.

Any comments are appreciated.

Thanks.

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1 Answer 1

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The answer depends on one's definition of "family of sets". There are basically two possibilities:

  1. A family of sets $(a_i)_{i \in I}$ is a mapping $f:I \to A$;
  2. A family of sets $\{a_i\}_{i \in I}$ is the image of a mapping $f: I \to A$.

(Remark: Both notations (round vs. curly braces) are seen for both concepts; I distinguish them notationally for clarity.)

The difference is in whether we consider $a_i$ to uniquely determine $i$, when a set occurs multiple times in the family.


In the first case, it's easy: set-theoretically, we have $f = \{(i, a_i): i \in I\}$, and it's clearly bijective with $I$ (projection to first coordinate). So in this case, $(a_i)_{i \in I}$ is countable precisely when $I$ is.

In the second case, it's more difficult. Countability of $I$ means that $\{a_i\}_{i \in I}$ is finite or countable. But even if $I$ is uncountable, it may be that $\{a_i\}_{i \in I}$ is countable.


So in either case, because of our definition of "family of sets" as a set, we can apply the definition of countability for sets. But in only one of these cases this can be translated to a definition in terms of $I$.

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So basically you say that in both of the cases the fact of being countable is a consequence of the definition for ordinary sets? I mean this implies that there's no need for a definition of countability over families but rather this depends just on the choice of either of the two alternatives for the definition of family of sets. right? –  Daniela Diaz Oct 10 '13 at 11:21
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Yes. A family of sets is itself a set, so the ordinary definition applies. What it amounts to depends on the specific definition (allowing repetition or not). –  Lord_Farin Oct 10 '13 at 11:28

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