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Math 185 Topology Worksheet Dave Penneys Summer 2009
1 Open and Closed Sets
Recall the following definitions from class: Definition 1.1. The ε-ball (or ε-neighborhood) around z 0 ∈ C is Bε(z 0 ) = {z ∈ C
|z − z 0 | < ε}^. The deleted ε-ball (or deleted ε-neighborhood) around z 0 ∈ C is Bε(z 0 ) \ {z 0 } =
z ∈ C
0 < |z − z 0 | < ε
Definition 1.2. Let S ⊂ C. A point z 0 ∈ C is called (1) an interior point of S if there is an ε > 0 such that Bε(z 0 ) ⊂ S, (2) an exterior point of S if there is an ε > 0 such that Bε(z 0 ) ⊂ Sc, and (3) a boundary point of S if z 0 is neither an interior point nor an exterior point of S.
Exercise 1.3. Show that z 0 is a boundary point of S if and only if for all ε > 0, Bε(z 0 ) ∩ S 6 = ∅ 6 = Bε(z 0 ) ∩ Sc. Definition 1.4. Suppose S ⊂ C. We define (1) the boundary of S as ∂S = {z ∈ C
z is a boundary point of S}, (2) the interior S as int(S) = S \ ∂S, and (3) the closure of S as S = S ∪ ∂S.
Exercise 1.5. Show that ∂S = ∂Sc. Definition 1.6. Define the exterior of S as ext(S) = Sc^ \ ∂S. Exercise 1.7. Show that ext(S) = int(Sc). Definition 1.8. S ⊂ C is called (1) open if S = int(S), and (2) closed if S = S.
Exercise 1.9. Show that S is open if and only if Sc^ is closed. Hint: Use Exercise 1.5 and DeMorgan’s Laws.
Exercise 1.10. Show S = S. Hint: Use Exercises 1.7 and 1.9.
Exercise 1.11. True or False? (1) (S)c^ = Sc. (2) int(S)c^ = int(Sc).
Exercise 1.12. Show that (1) S = (int(Sc))c, and (2) int(S) = (Sc)c.
Definition 1.13. z 0 is an accumulation point of S if each deleted ε-neighborhood of z 0 intersects S, i.e. for all ε > 0, ( Bε(z 0 ) \ {z 0 }
∩ S 6 = ∅.
Exercise 1.14. Find a set with (1) no accumulation points, (2) one accumulation point, and (3) n accumulation points where n ∈ N.
2 Compactness
Theorem 2.1 (Hiene-Borel). The following are equivalent for a subset S of R or C: (1) S is closed and bounded. (2) Every open cover of S has a finite subcover, i.e., if
Ui
i ∈ I
is some collection of open sets such that S ⊂
i∈I
Ui,
then there is an n ∈ N such that S ⊂
Ui 1 ∪ · · · ∪ Uin
Definition 2.2. A subset S of R or C is called compact if one (both) of the conditions in 2.1 hold. Exercise 2.3. Use the second condition in 2.1 to show that the continuous image of a compact set is compact, i.e., if S ⊂ C is compact and f : C → C is continuous, then f (S) is compact. Hint: Note that f is continuous if and only if f −^1 (U ) is open for every open U ⊂ C.
Exercise 3.4. Suppose γ : [a, b] → C is a path. Show ˜γ : [−b, −a] → C is a path where ˜γ(t) = γ(−t). We usually denote ˜γ = −γ. Definition 3.5. A path γ is called piecewise linear if γ is the concatenation of finitely many linear paths. Exercise 3.6. Suppose γ : [a, b] → U is a path where U ⊂ C is an open set. Show that there is a piecewise linear path ˜γ : [c, d] → U such that ˜γ(c) = γ(a) and ˜γ(d) = γ(b). Hint: Use the ε-Collar Theorem 2.7.
Definition 3.7. A path γ : [a, b] → C is called horizontal if γ is linear and Im(γ(a)) = Im(γ(b)), vertical if γ is linear and Re(γ(a)) = Re(γ(b)), and rectangular if γ is the concatenation of finitely many horizontal and vertical paths. Exercise 3.8. Suppose γ : [a, b] → U is a path where U ⊂ C is an open set. Show that there is a rectangular path ˜γ : [c, d] → U such that ˜γ(c) = γ(a) and γ˜(d) = γ(b). Hint: Use the ε-Collar Theorem 2.7.
4 (Path) Connected and Simply Connected Sets
Definition 4.1. A subset S of C is called disconnected if there are nonempty open sets U, V ⊂ C such that (1) U ∩ V = ∅, (2) S ∩ U 6 = ∅ 6 = S ∩ V , and (3) S ⊂ U ∪ V. A subset of C which is not disconnected is called connected. Exercise 4.2. Show that ∅ is connected. Definition 4.3. A subset S of C is called path connected if for all z 0 , z 1 ∈ S, there is a path γ : [a, b] → S such that γ(a) = z 0 and γ(b) = z 1. Exercise 4.4. Show that path connectedness implies connectedness Example 4.5. Define Γ : (0, ∞) → C by
Γ(t) = t + i sin
(π t
Let S = im(Γ) ∪ { 0 }. Then S is connected, but not path connected. There is no path γ : [a, b] → S with γ(a) = 0 and γ(b) = 1.
Theorem 4.6. Suppose U ⊂ C is open. Then U is connected if and only if U is path connected. Definition 4.7. A subset D ⊂ C is called a domain if D is open and connected. Definition 4.8. A subset S of C is called bounded if there is an N ∈ N such that S ⊂ BN (0). Theorem 4.9. Let γ be a simple closed path in C. Then im(γ)c^ is disconnected. Moreover, there are connected open sets U, V such that (1) U ∩ V = ∅, (2) im(γ)c^ = U ∪ V , and (3) U is bounded and V is not bounded. We denote U = ins(γ) and V = out(γ), the inside and the outside of γ. Definition 4.10. A domain D is called simply connected if for every simple closed path γ : [a, b] → D, we have ins(D) ∩ Dc^ = ∅. Exercise 4.11. Show that C \ { 0 } is a domain, but not simply connected. Exercise 4.12. Show that the continuous image of a connected set is continuous.
5 (Local) Uniform Convergence
Definition 5.1. Suppose (fn) is a sequence of functions fn : → C for some domain D, and suppose f : D → C. (1) If S ⊂ D, we say fn → f uniformly on S if for all ε > 0, there is an N ∈ N such that |fn(z) − f (z)| < ε whenever n ≥ N and z ∈ S. (2) We say fn → f locally uniformly if fn → f uniformly on every compact subset of D.
Fact 5.2. If fn → f uniformly, then fn → f locally uniformly. Exercise 5.3. Show that if D = B 1 (1), f (z) = (1 − z)−^1 , and
fn(z) =
∑^ n j=
zj^ ,
then fn → f locally uniformly on D, but not uniformly.
Hint: Note that (^1) −^1 z =
(n∑− 1
j=
zj
n 1 − z. Theorem 5.4. Suppose fn → f locally uniformly on some domain D. (1) If fn is continuous for all n ∈ N, then so is f.
