12.9. EXERCISES 233

12.9 Exercises1. Let U = {(x,y,z) such that z > 0}. Determine whether U is open, closed or neither.

2. Let U = {(x,y,z) such that z≥ 0} . Determine whether U is open, closed or neither.

3. Let U ={(x,y,z) such that

√x2 + y2 + z2 < 1

}. Determine whether U is open, closed

or neither.

4. Let U ={(x,y,z) such that

√x2 + y2 + z2 ≤ 1

}. Determine whether U is open, closed

or neither.

5. Show carefully that Rn is both open and closed.

6. Show that every open set in Rn is the union of open balls contained in it.

7. Show the intersection of any two open sets is an open set.

8. If S is a nonempty subset of Rp, a point x is said to be a limit point of S if B(x,r)contains infinitely many points of S for each r > 0. Show this is equivalent to sayingthat B(x,r) contains a point of S different than x for each r > 0.

9. Closed sets were defined to be those sets which are complements of open sets. Showthat a set is closed if and only if it contains all its limit points.