Topology if the Complex Plane

Reading

Section 1.4, part II

Problems

Practice problems:

1. True or False: Union of two open sets is open.
2. True or False: Intersection of two open sets is open.
3. True or False: Union of two closed sets is closed.
4. True or False: Intersection of two closed sets is closed.
5. What about questions 1-4 but for infinitely many sets rather than just 2?
6. An “open rectangle” \((a,b)\times (c,d)\) is defined as all complex numbers whose real part is between \(a\) and \(b\) and whose imaginary part is between \(c\) and \(d\). Show that the open rectangle is in fact an open set.
7. Show that the “closed rectangle” is in fact a closed set.
8. True or False: A set can be both open and closed at the same time.
9. Suppose \(U\) is an open set, \(z_n\) a sequence, \(z_n\to z\) and \(z\in U\). Show that there is an \(N\) such that for all \(n\geq N\) we have \(z_n\in U\).
10. Show that an open set does not contain any of its boundary points.
11. Show that a closed set contains all of its boundary points.
12. Show that a single point is a closed set.
13. True or false: If a set contains all of its boundary points, then it is a closed set.
14. True or false: If a set contains none of its boundary points, then it is an open set.
15. Suppose \(f\colon\mathbb{C}\to\mathbb{C}\) is continuous, and \(U\subset\mathbb{C}\) is set. Recall the definition of the set \(f^{-1}(U) = \left\{ x\mid f(x)\in U\right\}\), which makes sense regardless of whether \(f\) is an invertible function. Show that if \(U\) is an open set then \(f^{-1}(U)\) is an open set. (Use \(\epsilon-\delta\) definition of continuity)
16. Show that if \(f\) is continuous and \(K\) is a closed set then \(f^{-1}(K)\) is closed.
17. Find a continuous function \(f\colon\mathbb{C}\to\mathbb{C}\) and a set \(U\) that is open but such that \(f(U)\) is not open.
18. (Challenge) Find a continuous function \(f\colon\mathbb{C}\to\mathbb{C}\) and a set \(K\) that is closed but such that \(f(K)\) is not closed. (It will have to be an unbounded set)
19. Show that if \(K\) is a compact set and \(F\) is a closed set, then \(K\cap F\) is compact.
20. Show that if \(f\) is continuous and \(K\) is a compact set then \(f(K)\) is a compact set. (use definition via sequences)

Topics to know

0. Notions of boundary points and limit points.
1. Open disc of radius \(r\) around a point. Picture.
   • Also “closed disc”.
2. Notion of open set.
   • For each of its points, the set contains an open disc centered at each that point.
   • The open disc itself is an open set. This is not obvious.
   • What is the largest open set? The smallest?
   • What are the open sets in the real line?
3. Notion of a closed set.
   • Closed sets are the complements of open sets.
   • What is the smallest/largest closed set?
   • The closed disc is a closed set.
4. A set is closed iff the limit of every convergent sequence from the set is also in the set.
   • The closed disc is a closed set, proof by thinking of the characterization via sequences.
5. Closed and bounded sets are called compact.
   • Set is compact if and only if every sequence from the set has a subsequence converging within the set.
6. Polygonally-connected sets.
   • Open and polygonally-connected sets are called regions.
7. Continuous functions.
   • Definition via \(\epsilon-\delta\).
   • Definition via sequences.