From subcompact to domain representable

The Banach–Mazur game and domain theory

“…In 2003 K. Martin [8] showed that, if a space is domain representable, then the player α has a winning strategy in the strong Choquet game. In 2005 W. Fleissner and L. Yengulalp [4] showed that it is sufficient that a space is countably domain representable. Now, we shall show that the property of being countably domain representable is necessary.…”

“…Proof. By Theorem 4.3 (3) in [4] (see also [8]) it suffices to prove that if X is Choquet complete, then X is F-Y countably domain representable. Assume that X is Choquet complete.…”

“…W. Fleissner and L. Yengulalp in [4] introduced a notion of a πdomain representable space, this is analogous to the notion of a domain representable.…”

When Cp(X) is domain representable

“…If the converse of Lemma 3.4 were true, then that converse, together with Lemma 3.5 and Tkachuk's Theorem, would give a proof of Theorem 4.1. However, the converse of Lemma 3.4 is false, [9]. Lemma 3.5.…”

“…Suppose that (Q, , B) represents X . Then the ideal completion of (Q, ), denoted Idl(Q), is a continuous dcpo, [1, Proposition 2.2.22], and X is homeomorphic to max(Idl(Q)), [9]. This method is used in [4] to show that subcompactness implies domain representability.…”