I was trying to examine examples of topologies as a self learning project (you can see the list here), and these questions arose:
- 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.
- Related to above: Can product topology on two finite topologies ever be infinite?
- 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?
- 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?