TD Property and Spatial Sublocales

As one of main backgrounds of locale theory, topologies have close connections with locales. But locales have other backgrounds such as algebra, mathematical logic, etc. So there are many differences between locales and topologies. Spatiality is an important localic property to investigate the connections between locales and topologies. The TD property is a special separation property which plays an important role in this kind of investigations. Just as it will be proved in this paper, the TD property often appears as the lowest requirement for many topological spaces such that they can be described with localic properties and vice versa. In this paper, we show these special properties of the TD axiom and investigate some other interesting and important problems of TD-spatiality of locales.