  We study $G_\delta$ subspaces of continuous dcpos, which we call
domain-complete spaces, and $G_\delta$ subspaces of locally compact sober
spaces, which we call LCS-complete spaces. Those include all locally compact
sober spaces-in particular, all continuous dcpos-, all topologically complete
spaces in the sense of \v{C}ech, and all quasi-Polish spaces-in particular, all
Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly
Choquet-complete, completely Baire, and $\odot$-consonant-in particular,
consonant; that the countably-based LCS-complete (resp., domain-complete)
spaces are the quasi-Polish spaces exactly; and that the metrizable
LCS-complete (resp., domain-complete) spaces are the completely metrizable
spaces. We include two applications: on LCS-complete spaces, all continuous
valuations extend to measures, and sublinear previsions form a space
homeomorphic to the convex Hoare powerdomain of the space of continuous
valuations.
