TITLE: Product of topological spaces and Kuratowski closure axioms
QUESTION [5 upvotes]: (Both finite and infinite) product of topological spaces are often expressed through open sets.
Can it be expressed instead in terms of Kuratowski closure axioms (directly in terms of Kuratowski axioms, not through the isomorphism between the structure defined by Kuratowski closure axioms and and sets of open or closed sets)?

REPLY [2 votes]: About a month ago I was one of organizers of a conference dedicated to the 120-th anniversary of  Kuratowski. There were many talks devoted to his contribution in mathematics,  but no one related to the question you asked. I am afraid that a direct application of the closure operator to product topology definition is problematic, because this topology is defined by its base of open sets and (as far as I know) there is no direct counterpart of a base for closed set, only dual, that is via complements. 
PS. You also may search about your question in the topology book by Kuratowski.