A NOTE ON THE MARTIN TOPOLOGY OF THE SPACE OF THE FORMAL BALLS

Let (X,d) a metric space and BX = X × denote the partially ordered set of generalized formal balls in X. We investigate the relations between the Martin topology and the product topology of certain topologies of X and the Sorgenfrey line. We give a condition that the Martin topology coincides with the product topology of a metric topology and the Sorgenfrey topology, and consider on the conditions that the Martin topology is homeomorphic to the product topology of a metric topology and the Sorgenfrey topology. We also show that the space of formal balls on with the Martin topology is homeomorphic to the square of the Sorgenfrey lines. 1 Introduction Basic tools of topological approaches to domain theory are functions such as a qausi-metric and a (weak) partial metric. The Scott topology and the Lawson topology are known as the fundamental topologies related to the order structures in posets. Several authors investigated the relations between order structures, metric-like functions above, the Scott topology and the Lawson topology (cf. (4)). K. Martin (7) introduced a notion of a measurement in a domain to describe a quanti- tative statements on programs, and P. Waszkiewicz (10) discussed on the relations between the measurements and the partial metrics due to S. G. Matthews (8) on continuous posets. K. Martin also showed that every measurement induces a topology that we call the Martin topology. It is shown that the Martin topology has a clopen base, and it is stronger than the Lawson topology. However, a few facts are known about the Martin topology. Let (X, d) be a metric space. Then, an element of B + X = X ×(0, +∞) is called a formal ball. We induce a partial orderon B + X as (x, r) � (y, s )i fd(x, y) ≤ r − s. The notion of formal balls is introduced by Weihrauch and Schreiber to represent a metric space in a domain as a computational model (11). Several authors sudied the poset of formal balls as an approximating structure of a metric space (1, 2, 5, 6). Recently, Tsuiki-Hattori (9) introduced formal balls with negative radiuses and study the partially ordered set BX = X × R with an order relation which is similar to B + X. An element of BX is called a generalized formal ball. The sets BX obviously has the Lawson topology as a poset. It is easy to see that the relative Lawson topology on every slice X ×{ t }⊂ BX (t ∈ R) is homeomorphic to the metric topology of X and every slice {x }× R ⊂ BX (x ∈ X) is homeomorphic to the usual real line R. In this direction, Tsuiki- Hattori considered the differences, or coincidences of the Lawson topology and the product topology of X and R on BX. The Martin topology on the space of formal balls seems to be more complicated, because the relative Martin topology on every slice X ×{ t }⊂ BX (t ∈ R) is a discrete space, which is not homeomorphic to the metric topology of X in general, and every slice {x }× R ⊂ BX (x ∈ X) is homeomorphic to the Sorgenfrey line. In the present note, we will consider