Important classes of topological
spaces have topologies which are induced by a generating collection of closed subsets;
typical examples are k-spaces, sequential spaces with unique sequential limits,
and lattices with the Birkhoff interval topology. This paper proceeds by
axiomatizing this construction—a set with a specified generating collection
of closed subsets is called a “hypotopological space.” The Birkhoff interval
topology is then studied in these terms. A natural embedding of hypotopological
spaces in conditionally complete, atomic, distributive lattices with Birkhoff
interval topology is derived. This embedding is used to show that lattices with
Birkhoff interval topology have the same nontrivial subspace and product
properties as k-spaces and sequential spaces. In particular, we answer in the
negative a question first raised by Birkhoff, namely, whether the Birkhoff
interval topology is preserved under the formation of the product of two
lattices.
