On a Smyth Conjecture
by
Michael A. Bukatin
Brandeis University
Coauthors: Svetlana Yu. Shorina (Moscow State University)

The set I_x={y in A | {x, y} is unbounded} is an observable continuous representation of negative information about x in A for a weakly Hausdorff continuous dcpo A with the Scott topology. When A is not weakly Hausdorff, the largest continuous approximation of  I_x is represented by J_x={y in A | x in  int(I_y)}, and the largest observable continuous representation of I is represented by J'_x=int(J_x).

Mike Smyth conjectured that J or J' is closely related to the least symmetric closed tolerance on A. In this paper we establish that, indeed, {<x, y>| y in J'_x} is the complement of this tolerance.

We also establish a relationship between this tolerance and lower bounds of relaxed metrics on A.

Date received: March 26, 1999