Fréchet space
A product of countably many Fréchet spaces is always once again a Fréchet space. However, an arbitrary product of Fréchet spaces will be a Fréchet space if and only if all for at most countably many of them are trivial (that is, have dimension 0). Consequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis). So for example, if formula_68 is any set and formula_1 is any non-trivial Fréchet space (such as formula_70 for instance), then the product formula_71 is a Fréchet space if and only if formula_72 is a countable set.