In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces, then a P-bounded space is one in which every subspace with property P has compact closure. Every compact space is ω-bounded, and every ω-bounded space is countably compact. The long line is ω-bounded but not compact. The bagpipe theorem describes the ω-bounded surfaces.
References Juhász, Istvan; van Mill, Jan; Weiss, William (2013), "Variations on ω-boundedness", Israel Journal of Mathematics, 194 (2): 745–766, doi:10.1007/s11856-012-0062-8, MR 3047090