This is very misleading. First of all openess is relative to the space you're in.

For example, you would agree with me that as a metric space (or a topological space...whichever is more up your ally) (with the usual metric/topology) considers closed. But, if you consider as being a subspace in it's own right, then it is open in itself. You can't be open. You have to be open in a specific space.

I'm being too finicky. You obviously mean these to be subsets of with the usual topology.

- Every space is open and closed within itself, it's the definition of a topology! Or, if you're just doing metric spaces note that it's closed since any limit point would have to be a point of the space and thus automatically in it. Also, any point is an interior point since any neighborhood is contained within the space! It's perfect for the same reason . But, too for if then taking any neighorhood of will hit another point of . It's obviously not bounded under the usual metric.

. I'll assume you mean . It's closed since given any we have that can contain at most one point, and so isn't a limit point. It follows that has no limit points and so trivially contains them. is not open for the easy reason that is connected and (among other reasons). It's not perfect since it is non-empty while it's set of limit points is (or because every pefect subset of is uncountable). Finally, it is also clearly nt bounded.