- What is the set of limit points of in ?
- Describe all the closed subsets of
- Describe all the compact subsets of
- Describe all the connected subsets of

- 0 is the only limit point. It is a limit point since as and all other points are isolated, so cannot be limit points.
- A subset is closed if it is finite or contains 0 (or both). To see this, recall that is closed in if and only if it contains all its limit points. A limit point of must be a limit point of so the only possibilities are that has no limit points or it contains In the latter case it is closed and the former means that 0 is either not in and is not a limit point of which therefore must be finite.
- Since contains its only limit point in it is closed and bounded in so it is compact by the Heine-Borel Theorem. Thus the compact subsets of are just the closed sets as described above, since as shown in class a subset of a compact space is compact if and only if it is closed.
- Let be a connected set. If is one of the isolated points of then is a decomposition with and so and Thus by the definition of connectedness and Thus the only possibility is that consists of any one point of and these sets are trivially connected since in any decompostion with one of or must be empty.