Igor Zelenko, Fall 2009
1
Homework Assignment 7 in Topology I, MATH636
due to Oct 26, 2009
(To get 100 it is enough to solve all problems except 3b.
If you will give also a solution for problem 3b, you will get up to 25 extra points )
1.
a. Show that the one-point compactification of the set of natural number N is
homeomorphic to the subspace {0} ∪ { n1 , n ∈ N} of R.
b. Let A be either open or closed subset of a locally compact Hausdorff space.
Show that A is also locally compact Hausdorff space (in the subspace topology).
c. Show that every locally compact Hausdorff space is regular.
2. Let Y be the compactification of the open interval (0, 1) induced by the embedding f : (0, 1) ,→ R2 defined by f (x) = (x, sin x1 ). Actually Y is the topologist
sine curve.
a. Show that the function g : (0, 1) → R given by g(x) = cos x1 can not be
extended continuously to Y .
b. Define an embedding h : (0, 1) ,→ R3 such that functions x, sin x1 , and cos x1
can be extended continuously to the compactification induced by h.
3.
a. Let X be completely regular. Show that X is connected if and only if its
Stone-Čech compactification β(X) is connected (Hint: If X is not connected
then there exists a continuous function from X to a dicrete set {0, 1}).
b. ( bonus of 25 points) Solve problem 3, p. 35 in the text.
4.
a. Show that if X has the discrete topology then X is paracompact.
b. Prove that the product of a paracompact space and a compact space is
paracompact (Hint: Use the tube lemma)
5. Solve problem 2, p. 39 in the text.