# Countability of Topologies and Product Topology

I was trying to examine examples of topologies as a self learning project (you can see the list here), and these questions arose:

1. I couldn't spot a product topology where any set other than basis element is open in it - that is, a product topology where all open sets are not basis elements of the form U x V, where U and V are open in their respective spaces.
2. Related to above: Can product topology on two finite topologies ever be infinite?
3. Product of countably infinite sets is uncountable. Can I safely assume that product topology on a countable collection of spaces with countably infinite topologies is uncountable?
4. Related to above: What can we say about countability of product topology on product of countable collection of finite sets (and hence topologies on each space must be finite)? Am I right in guessing it must be countable?

1. The open unit circle in $$\mathbb{R}^2$$ is an example, or $$\{(x,y): x \neq y\}$$. The plane has the product topology wrt the usual topology on $$\mathbb{R}$$. These sets are open but not Cartesian products of two sets.
4. It can and often will be uncountable, as in the case of the Cantor cube $$\{0,1\}^\mathbb{N}$$. e.g.