Overview and comparison of localic and fixed-basis topological products

This paper studies localic products, traditional topological products, and L-topological products, and gives a complete outline of the localic product. Comparisons of localic and L-topological products are generally absent in the literature, and this paper answers longstanding open questions in that area as well as provides a complete proof of the classical comparison theorem for localic and traditional topological products. This paper contributes several L-valued comparison theorems, one of which states: the localic and L-topological products of L-topologies are order isomorphic if and only if the localic product is L-spatial, providing L is itself spatial and the family of L-topological spaces is ''prime separated''. These last two conditions always hold in the traditional setting, capturing the traditional comparison theorem as a special case, and the prime separation condition is satisfied by important lattice-valued examples that include the fuzzy real line and the fuzzy unit interval for L any complete Boolean algebra and the alternative fuzzy real line and fuzzy unit interval for L any (semi)frame. Separation conditions help control the ''sloppy'' behavior of the L-topological product when |L|>2, and several separation conditions are studied in this context; and it should be noted that localic products have a point-free version of the ''product'' separation condition considered in this paper. The traditional comparison theorem is carefully proved both to fill gaps in the extant literature and to motivate the L-valued comparison theorem quoted above and reveal the special role played by cross sums of prime (L-)open subsets. En route, characterizations are given of prime L-open subsets of certain L-products, which in turn yield characterizations of prime open and irreducible closed subsets of traditional product spaces.