This is homework 3, due Monday, October 17, at the beginning of lecture.
- Recall the Baire category theorem: The intersection of countably many dense open sets in a complete metric space is dense. Recall that a set is one that can be written as the intersection of countably many open sets.
Show that in a complete metric space, the intersection of countably many dense sets is again a dense set. Is the union of two sets again a set?
Show that is not a subset of .
Since is countable, we can enumerate it as . For let
Note that each is open, that for each , that , and that (for each ) the sum of the lengths of the intervals that make up is at most .
Does this mean that ? If yes, please provide a proof. If not, describe as concretely as possible an irrational number that belongs to this intersection.
- Recall that a nowhere dense set is a set whose closure has empty interior: . A set is meager (or of the first category) iff it is the union of countably many nowhere dense sets. Given sets and in , we say that iff their symmetric difference is meager. For example, the Cantor set is nowhere dense.
Show that is an equivalence relation.
Show that if is meager, then any subset of is also meager. Show that the union of countably many meager sets is again meager.
Show that the Baire category theorem implies that any nonempty open subset of is non-meager.
Show that if is open, or closed, or a set, or a set (a countable union of closed sets), then there is an open set such that .
- The Cantor-Bendixson derivative of a closed set is defined by
Since is closed, . We can iterate this operation, and form , Note we have
We can go even further, by letting and then continuing, by setting , etc.
Give examples of closed subsets of such that but , or but , or or but . Can be the first empty “derivative”? How about ?
- Check that if and are compact, then so is as a subset of .
- Show that if is a subset of that is both open and closed, and , then it must be the case that .