They can also be finite, countably infinite, or uncountably infinite. The union of countably many countable sets is countable. A countable set is either a finite set or a countably infinite set. AC suffices to prove that the union of countably many countable sets is countable. What about infinite subsets of countably infinite sets? The Eisenstein integers are a countably infinite set. Some authors use countable set to mean "countably infinite" alone. The number of permitted values is either finite or countably infinite. The set of possible output values may be finite or countably infinite. In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension. In particular, every infinite subset of a countably infinite set is countably infinite. Observe that in that theorem can be replaced with any countably infinite set. The situation is different for countably infinite compact Hausdorff spaces. There are countably infinite souls in the universe. The dimensionality of "F" is countably infinite.