Igor Zelenko, Fall 2009
1
Homework Assignment 4 in Topology I, MATH636
due to Sept 28, 2009
1. Solve problem 4, p. 14 in the text.
2. Solve problem 6, p. 14 in the text.
3. Let f, g : X → Y be continuous functions and Y be a Hausdorff space. Assume that there exists
a dense subset A of X such that f (x) = g(x) for any x ∈ A. Prove that f (x) = g(x) for any
x ∈ X.
4. Let (X, ≤) be a totally (linearly) ordered set. Prove that X is compact in the order topology if
and only if every nonempty subset of X has a greatest lower bound and a least upper bound.
5. Let {Kα }α∈A be a collection of compact subsets of a Hausdorff space X which is closed with
\
respect to finite intersections. Let K =
Kα .
α∈A
a. Prove that K is compact.
b. Suppose that W is an open subset of X such that K ⊂ W . Prove that Kα ⊂ W for some
α ∈ A.
c. Prove that if Kα is connected for each α ∈ A, then K is connected.
6. Assume that X is a second countable topological space.
a. Prove that any open covering of X has countable subcover (a topological space satisfying
the last property is called a Lindelöf space; in other words, you are asked to prove that any
second countable space is Lindelöf);
b. Prove that X is compact if and only if any sequence in X has a convergent subsequence
(a topological space satisfying the last property is called sequentially compact; in other
words, you are asked to prove that for second countable topological spaces compactness is
equivalent to sequential compactness).
7. A space X is called countably compact if every countable open covering has a finite subcover.
a. Show that X is countably compact if and only if every countable family of closed subsets
having the finite intersection property has nonempty intersection.
b. If X is countably compact and Y is Hausdorff and second countable, then a continuous
bijection f : X → Y is a homeomorphism.