Pointfree Aspects of the T_d Axiom of Classical Topology

Abstract

Abstraction from the condition defining To-spaces leads to the following notion in an arbitrary frame L: a filter F in L is called slicing if it is prime and there exist a, b e L such that a £ F, b e F, and a is covered by 6. This paper deals with various aspects of these slicing filters. As a first step, we present several results about the original td condition. Next, concerning slicing filters, we show they are completely prime and characterize them in various ways. In addition, we prove for the frames £>X of open subsets of a space X that every slicing filter is an open neighbourhood filter U(x) and X is td iff every U(x) is slicing. Further, for Top_D and Prm_D the categories of td spaces and their continuous maps, and all frames and those homomorphisms whose associated spectral maps preserve the completely prime elements, respectively, we show that the usual contravariant functors between Top and Frm induce analogous functors here, providing a dual equivalence between Top_D and the subcategory of Prm_D given by the To-spatial frames (not coinciding with the spatial ones). In addition, we show that Top_D is mono-coreflective in a suitable subcategory of Top. Finally, we provide a comparison between Jo-separation and sobriety showing they may be viewed, in some sense, as mirror images of each other.