Infinite cartesian product of singletons

Question

Suppose you have infinitely many different singleton sets. If you took the infinite cartesian product of all your singleton sets, would the result be one giant ordered pair? Would it matter whether you had countably or uncountably infinite many singletons?

Answer 1

The result would be a set of one element: a fixed function, essentially. If the singletons are $\{a_i\}$ for $i \in I$, $I$ being some index set, then:

$$\prod_{i \in I} \{a_i\} = \{f\}$$ where $f: I \to \{a_i : i \in I\}$ is defined by $f(i) = a_i$.

So one element regardless of the size of $I$ (countable, finite, uncountable..)