From subcompact to domain representable

We introduce the property generalized subcompactness and prove that subcompactness implies generalized subcompactness and that generalized subcompactness implies domain representability. We develop a simplified characterization of domain representability. We present an extension X of Debs' space and prove that X is generalized subcompact but α does not have a stationary winning strategy in the Banach–Mazur game on X  . A fortiori, domain representability does not imply subcompactness. We investigate whether GδGδ subspaces of subcompact (generalized subcompact, domain representable) spaces are subcompact (generalized subcompact, domain representable). We show that Čech complete generalized ordered spaces are subcompact. We show that the union of two domain representable subspaces is domain representable, and that a locally domain representable space is domain representable.