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empty product
The empty product of numbers is the borderline case of product, where the number of factors is zero, i.e. the set of the factors is empty. The most usual examples are the following.

The zeroth power of a nonzero number: $a^{0}$

The factorial of 0: 0!

The prime factor presentation of unity, which has no prime factors
The value of the empty sum of numbers is equal to the additive identity number, 0. Similarly, the empty product of numbers is equal to the multiplicative identity number, 1.
Note. When considering the complex numbers as pairs of real numbers one often identifies the pairs $(x,\,0)$ and the reals $x$. In this sense one can think that the Cartesian product $\mathbb{R}\times\{0\}$ is equal to $\mathbb{R}$. This seems to mean the equation
$\mathbb{R}\times\mathbb{R}^{0}=\mathbb{R}^{{1+0}}=\mathbb{R}^{1}=\mathbb{R},$ 
although the associativity of Cartesian product is nowhere stated. Nevertheless, it is sometimes natural to define that the Cartesian product of an empty collection of sets equals to a set with one element; so it may mean that e.g. $\mathbb{R}^{0}=\{0\}.$
One can also consider empty products in categories. It follows directly from the definition that an object in a category is a product of an empty family of objects in the category if and only if it is a terminal object of the category. Sets are a special case of this: in the category of sets the singletons are the terminal objects, so the empty product exists and is a singleton.
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Comments
Empty product of sets
I am not sure whether I satisfied your idea.
Re: Empty product of sets
It looks good to me. I like the example about the real axis in the complex plane. It just seemed that your entry would be a good place to mention this sometimes useful convention.
Is a^0 an empty product or not?
a^0 is an empty product for every a (not just nonzero a). After all, an empty set that contains no a=3’s is the same as an empty set that contains no a=0’s.