> Granted, infinite sets like aleph-0 and -1 are typically considered both open and closed This statement is meaningless. It makes no sense to speak of whethe

> Granted, infinite sets like aleph-0 and -1 are typically considered both open and closed This statement is meaningless. It makes no sense to speak of whether a set is open or closed outside the context of a topology. Also, like... in any topological space of the sort we usually deal with (specifically: T_1 spaces), any open set must be infinite, as any finite set is closed? So I'm really not sure what you're thinking of here. (And in something like Euclidean space -- or a manifold -- any open set must be uncountably infinite...) (Also, as others have already mentioned, "closed" here doesn't mean "closed" in the sense of point-set topology; "closed manifold" means a compact manifold [without boundary, if manifolds with boundary are being considered]. Yes, this is a bit confusing at first.)