Notes On The Scott Topology Of Product-domain And The Relevant Problem

In computer science,topology is used to explain approximate states of information and the Scott topology on Domain is used oftenly.A natural problem is:for any family of dcpos {Li}i∈I, whether the Scott topology of ∏(Li) keeps the same with the product of the Scott topologies?Through a exam-ple.we proves there are a family of dcpos {Lt}i∈I ,and every σ(Li) is continuous lattice.but is with least element, we proved that the conclusion is established.Meanwhile, way-below relation is the important tool describing the approximate states of information in terms of order. We got some results of the way-below relation of continuous function.On the other hand ,as for the problem of how to measure the approximate degree of information, Keye Martin brought forward new concepts such as measurement,induce Scott topology and u-topology. In the paper,we introduced the ↑μ-topology which is the popularization of u-topology and some basic properties of ↑μ-topology are studied.We showed that under the condition of stable partial metric space,the partial metric topology agreed with the↑μ-topology.