Homework 7. Due Friday, November 27.
1. Show that the unit circle C = {(x, y) ∈ R2 : x2 + y 2 = 1} is
connected. Hint: Use polar coordinates and Theorem 2.15.
2. Let U = {(x, y) ∈ R2 : xy 6= 0}. How many connected components
does this set have? Hint: One of the components is (0, ∞) × (0, ∞).
3. Show that there is no surjective continuous function f : A → B, if
A = (0, 3) ∪ (3, 6),
B = (0, 1) ∪ (1, 2) ∪ (2, 3).
Hint: Look at the restriction of f on each subinterval. You will need
to use Theorem 2.11 and also some parts of Theorem 2.15.
4. Let (X, T ) be a topological space and suppose A1 , A2 , . . . , An are
connected subsets of X such that Ak ∩ Ak+1 is nonempty for each k.
Show that the union of these sets is connected. Hint: Use induction.