The category of topological spaces and open maps does not have products

TL;DR

This paper demonstrates that the category of topological spaces with open maps, as well as certain algebraic categories, lack binary products or coproducts, resolving longstanding open problems in category theory.

Contribution

It proves that the category of topological spaces with open maps does not have binary products, addressing the Esakia problem negatively.

Findings

Category of topological spaces with open maps lacks binary products

Categories of complete Heyting and closure algebras lack binary coproducts

Resolves the Esakia problem in the negative

Abstract

We prove that the category of topological spaces and open maps does not have binary products, thus resolving the Esakia problem in the negative. We also prove that the categories of complete Heyting algebras and complete closure algebras do not have binary coproducts.