Topology Qualifying Exam
Topics Covered
MAT 661
Point Set Topology
Topological Spaces: Topologies, neighborhoods, basis, closure operations, continuity,
product and quotient topologies.
Topological Properties: Separation axioms, compactness, local compact
Topology Qualifying Exam
August 23, 2010
Do all of the following problems.
1. Let X be a topological space.
(a) (5 points) Dene what it means to say that X is regular.
(b) (10 points) Give a complete proof that a product of two regular spaces is
regular.
Topology Qualifying Exam
January 9, 2009
Do all of the following problems each of which is worth 20 points.
1. Let X be a topological space.
(a) Dene what it means to say that X is compact.
(b) State the Tube Lemma.
(c) Prove the Tube Lemma.
2. Let X and
T OPOLOGY QUALIFYING EXAM
AUGUST 2009
There are six problems. Begin y our a nswer to any p roblem on a n ew p age in y our blue
book(s). Make a space between answers to separate p arts of a question to facilitate
grading. A ll a nswers must be j ustified
August 2008
Qualifying Examination
Topology
Problems 1-4 consist of true or false statements. Each statement is to be proved or
disproved with brief but complete reasoning. Provide definitions of all underlined,
italicized words and phrases. On page 2 fin
Topology Qualifying Exam
August 20, 2007
Do all of the following problems.
1. (20 points) Let X be a set.
(a) Dene what it means to say that the collection B of subsets of X is a basis for
a topology on X .
(b) Suppose now that X is endowed with a topolog