Give an example, if possible, of a denumerable collection of finite sets whose union is finite.
haha, it's a letter, well, a set containing a letter. doesn't matter what a is. all your sets have one element, and that is called a. you have a denumerable collection of 1-element sets (hence finite sets), whose union is a 1-element set (hence finite). if it bothers you so much, you can replaced a with whatever you want. like a number or something.
the sets were not required to be denumerable, the collection, or "number", of sets are to be. each set should be finite. i agree with FernandoRevilla's interpretation (he wasn't very rigorous in his definition to begin with, but i think he got the idea across). And i don't think it would work without making a lot of the sets (a countably infinite amount of them) the same set.
If you tell me "X is a denumerable collection of finite sets whose union is finite" then I take that as:
X is denumerable
Every member of X is finite
The union of X is finite
The word "collection" has no special meaning here other than 'set' unless otherwise specified.
In particular, it's a jump to regard "collection" as an indexing function unless that is specified.