\(RW\)-spaces and compactness of function spaces for L-domains (Q1873730)

If \(X\) is a topological space and \(D\) is a directed complete partially ordered set with the Scott topology, then the set \([X\to D]\) consisting of all continuous morphisms from \(X\) to \(D\) with the pointwise order is again a directed complete partially ordered set. In this paper the question is considered under which circumstances the function space obtained by supplying \([X\to D]\) with the Scott topology is compact. \textit{J. Liang} and \textit{K. Keimel} [Comput. Math. Appl. 38, 81-89 (1999; Zbl 0941.68080)] have shown that if \(X\) is a core compact topological space with property \(W\) and \(D\) is a compact continuous \(L\)-domain, then \([X\to D]\) is a compact continuous \(L\)-domain. Here, a weaker topological property, called \(RW\), is introduced, and the following results are proved. Theorem 1: Every continuous domain \(D\) which is compact with respect to the Lawson topology has property \(RW\) when topologized with the Scott topology. Theorem 2: An \(L\)-domain has property \(RW\) in its Scott topology if and only if it is compact with respect to its Lawson topology. Theorem 3: For a core compact topological space \(X\) the following are equivalent: (i) \(X\) has property \(RW\); (ii) for every compact continuous \(L\)-domain \(D\) the function space \([X\to D]\) with the Scott topology is compact. Theorem 4: For a continuous directed complete partially ordered set \(D\) with a least element, the following are equivalent: (i) for every core compact topological space \(X\) with property \(RW\) the function space \([X\to D]\) is a compact continuous directed complete partially ordered set; (ii) \(D\) is a compact continuous \(L\)-domain.