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Sep 9, 2022 at 8:27 comment added Leon Staresinic @WlodAA they are not, if they were, the complement would always be path connected. I am aware that some kind of Alexander duality should be lurking in the background but I am not able to rephrase the equivalence in homological terms.
Sep 9, 2022 at 0:00 comment added Tom Goodwillie It's not even necessary to assume that your open set is a union of the interiors of finitely many topological closed disks. This has to do with Alexander duality.
Sep 8, 2022 at 23:18 comment added Wlod AA Are these disks assumed to be pairwise disjoint?
Sep 8, 2022 at 20:39 history asked Leon Staresinic CC BY-SA 4.0