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My question is pretty much the title. I just find this notion a little weird. Say for example that I have the set $A$ that is uncountable. Can I say $A = \{ a_{i} \}_{i\in I}$? Thanks

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I guess I might add that the set $I$ should be uncountable also. – Jmaff Dec 22 '12 at 4:51
The set $I$ would have to be uncountable, because you can't have a surjection from a countable set to an uncountable one. – Trevor Wilson Dec 22 '12 at 4:52
up vote 8 down vote accepted

To say that there is an indexing $\{a_i\}_{i\in I}$ of $A$ by $I$ is the same as saying that there is a surjection $f: I \to A$. It's just a matter of notation: $a_i$ versus $f(i)$. The simplest example of a surjection $f:I \to A$ from some set $I$ onto $A$ is given by setting $I = A$ and letting $f$ be the identity function. This corresponds to the trivial indexing $\{a_i\}_{i\in A}$ of $A$ where $a_i = i$.

In some applications you might want $I$ to be an ordinal. For this it is a necessary and sufficient condition that a well-ordering of $A$ exists. In particular, if the Axiom of Choice holds then any set $A$ can be indexed by an ordinal.

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That helps, thanks. – Jmaff Dec 22 '12 at 5:09

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