Topology: Problems
1. Let S be any subset of the real numbers. Prove that int(S) is open. Prove that
S is open if and only if S = int(S).
2. Let S be any subset of real numbers. Define the closure of the set S as cl(S)
= S 
bd(S). Prove that cl(S) is a closed
set.
3. Let K be a compact set and U an open set containing K. Prove that there
exists an 
> 0 such that if k 
K then the interval (k - , k + ) is contained in U.
4. Prove that every closed subset of a compact set is compact.
5. Let S be any subset of real numbers. Call a set robust if it is equal to the closure of its interior.
Then
- Is every open set robust ? Is every closed set robust ?
- Is every robust set open ? Is every robust set closed ?
- Is every closed interval robust ? Is every robust set equal to a closed interval ?
- Make a conjecture to characterize all robust sets in the real line.
6. Let 
, 
, ... be perfect sets
and suppose that 
.. Is the intersection of all 
perfect ?
7. Give an example of non-empty, closed sets ... such that their intersection is empty.
8. Give an example of non-empty, closed sets 
... such that their union is open.
9. Give an example of open sets ... such that their union is closed and non-empty
10. Give an example of a totally disconnected subset S of the interval [0, 1] such that the closure
cl(S) = [0, 1].
11. What is the interior of the Cantor set ? What is the boundary of the Cantor set ?
12. Write the real line as the union of two totally disconnected sets.
13. Let , , , ... be closed sets such that their union is equal to the real line. Prove that at least
one of the sets must have non-empty interior. (Hint: Look at the
proof that perfect sets are uncountable).
(bgw)