Topology in Computer Science Problems

We pose the problem of whether every FS-domain is a retract of abifinite domain purely in terms of quasi-uniform spaces. 6.1 The problem and its historyEver since domains were introduced by Dana Scott [Sco70] and Yuri Er-shov [Ers75], a question in the centre of interest was to find suitable cartesianclosed categories of domains and the quest for cartesian closed categories aslarge as possible, so to be closed under various domain-theoretic construc-tions.Gordon Plotkin defined in [Plo76] SFP-domains and conjectured that thecategory of ω-algebraic SFP domains is the largest cartesian closed categoryof domains. This conjecture was confirmed by Mike Smyth in [Smy83].In [Jun88, Jun89], Achim Jung dropped the countability condition andshowed that there are two maximal cartesian closed categories of algebraicdomains, bifinite domains and L-domains. Moreover, he proved that everycartesian closed category of algebraic domains is contained in one of these.For certain applications in semantic the more general class of continuousdomains is considered rather than that of algebraic domains. Continuousdomains are exactly the retracts of algebraic domains. Moreover, givena cartesian closed category of algebraic domains, the category comprisingthe retract of its objects is cartesian closed again. As shown by AchimJung in [Jun90] the picture is similar to the algebraic case: There are ex-actly two maximal cartesian closed categories of continuous domains, that ofFS-domains and that of continuous L-domains, and every cartesian closedcategory of continuous domains is contained in one of these. ContinuousL-domains are exactly the retracts of L-domains, so it is a natural questionto ask whether the same is true for bifinite domains / FS-domains. Question. Is every FS-domain the retract of a bifinite domain?