Direct decompositions of lattices of continuous functions
Robert L. Blair, Claude W. Burrill · Proceedings of the American Mathematical Society · 1962
If X is a topological space and if K is a chain equipped with its order topology, then we denote by C(X, K) the lattice of all continuous functions from X to K. If X is the union of two disjoint open-andclosed subsets Xi and X2, then it is clear that C(X, K) is isomorphic to the direct product of the lattices C(X1, K) and C(X2, K). In Theorem 2 of [2], Kaplansky proves the following converse: