Structure of lattices characterized by validity of lattice-valued topological propositions

Topological properties of a space consisting of lattice-valued mappings, namely, topologies on a lattice, as well known, are certainly affected by ordered structures on the range. In this paper, some stronger inverse results will be proved. Some ordered structure on the range can be characterized by the presence of topological or analytical properties such as analytic characterizations of lattice-valued semicontinuous mappings, lattice-valued Hewitt–Marczewski–Pondiczery Theorem, etc. In fact, the ordered structure and the topological property are determined by each other.