T D 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. TheT _D 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, theT _D 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 theT _D axiom and investigate some other interesting and important problems ofT _D -spatiality of locales.Supported by the National Natural Science Foundation of China and the Science Foundation of the State Education Commision of China.Supported by the Fund for Excellent Young University Teachers of the State Educational Commission of China and theE _x -Oversea-Scholars Fund of the Educational Commission of China.