Math 4530 ā First Midterm Exam
2:55pmā4:10pm, Thursday 1st October 2009
āThe mathematician does not study pure mathematics because it is useful;
he studies it because he delights in it and he delights in it because it is beautiful.ā
Henri PoincarĀ“e.
Please answer all the questions and justify you answers. Use of calculators and other
electronic devices is not permitted. Notes and books may not be used. Please write you
name on every sheet you hand in. At the end of the exam you will be asked to number
your pages.
1. (a) Suppose f:XāYis a continuous function between topological spaces. Prove that if Xis
pathāconnected, then f(X) is a pathāconnected subset of Y.
(b) Use the result of part (a) to prove that the 2-sphere S2=ī(x, y , z)āR3|x2+y2+z2= 1ī
(with the subspace topology inherited from R3) is path connected. (3 + 3 pts)
2. (a) Define the subspace topology on a subset Aof a topological space X.
(b) Verify that the subspace topology satisfies the axioms for a topological space. (3 + 3 pts)
3. (a) What does it mean to say a subset Aof a topological space Xis connected?
(b) Give a proof or counterexample for each of the following statements.
i. If Aand Bare connected subsets of a topological space and Aā©B6=ā
, then AāŖBis
connected.
ii. If Aand Bare connected subsets of a topological space, then Aā©Bis connected.
(2 + 3 + 3 pts)
4. Which of the following spaces are compact?
(a) The subset ī(x, y, z)āR3|x2+y2+z2= 1 and x > 0īof R3(with the usual topology).
(b) The set {1,2,...,n}with the discrete topology.
(c) The subset [0,1] of Rwhere Rhas the topology that has basis {[a, b)|a, b āR, a < b }.
(d) The subset Zof Rwhere Rhas the topology in which sets are open when they are empty or
have countable complement.
(2 + 2 + 2 + 2 pts)
(Total = 28 pts)
TRR, September 2009