Math 434/Math 542
Test 2
Carter
Fall 2012
I recommend an occasional cup of warm milk in the late evening while reading some light
fiction. Usually one minute and thirty seconds in the microwave at full power is sufficient.
Wait about 30 minutes after drinking the milk before brushing your teeth.
1. State the definition of a Hausdorff topological space.
2. Prove that points are closed in a Hausdorff space.
3. State the definition of a compact space.
4. Give an example of a compact space that is not homeomorphic to the product of finitely
many closed intervals.
5. Show that a compact Hausdorff space is regular.
6. Prove that a (Hausdorff) space Y is normal if and only if for any two closed setsA, B ⊂
Y there is a continuous function f : Y → [0, 1] such that
(a) f (a) = 0 ∀a ∈ A, and
(b) f (b) = 1 ∀b ∈ B.