Eksamen juni 2003

Matematik 3GT

Books, notes, calculators, and computers are allowed at this three hour written exam. You may write your answers in pencil.

Problem 1 (20 points)

Let S_{Ω} denote the smallest uncountable well-ordered set.

(1) Show that S_{Ω} contains a subset with the order type of the positive integers Z_{+}. [Hint: S_{Ω} does not have a largest element.]

(2) Find an element of S_{Ω} that does not have an immediate predecessor. [Hint: Use Z_{+} ⊂ S_{Ω}.]

Problem 2 (10 points)

If Z is a topological space and C ⊂ Z a subset, we define the boundary of C by the equation

∂C = \overline{C} ∩ \overline{(Z − C)}

Let X and Y be topological spaces, A a subset of X and B a subset of Y. Then A × B is a subset of X × Y. Show that

∂(A × B) = (∂A × \overline{B}) ∪ (\overline{A} × ∂B)

[Hint: (X × Y) − (A × B) = (X − A) × Y ∪ X × (Y − B)]

Problem 3 (20 points)

Let f: S^{1} → R be a continuous map of the circle to the real line.

(1) Show that there is a point x on the circle so that f(x) = f(−x). [Hint: The odd map g(x) = f(x) − f(−x) must take the value 0 at some point.]

(2) Is it possible to imbed the circle S^{1} in the real line R?

Problem 4 (40 points) Let X = Z_{+} be the set of positive integers with the discrete topology and β(X) its Stone–Čech compactification. We consider X as a subset of β(X). Let A be any subset of X ⊂ β(X). Let U be an open subset of β(X).

(1) Show that there is a continuous function F: β(X) → {0, 1} defined on the compactification such that F (A) = 0 and F (X − A) = 1. Deduce that \overline{A} and \overline{X − A} are disjoint where closures are taken in β(X).

(2) Show that β(X) − \overline{A} = \overline{X − A} and that \overline{A} is open and closed in β(X). [Hint: You may use without proof the general fact that \overline{C} − \overline{D} ⊂ \overline{C − D}.]

(3) Show that \overline{U} = \overline{U ∩ X} and that \overline{U} is open and closed in β(X).

(4) Show that the connected components of β(X) are one-point sets.

(THE END)