Exam 2: Math 6342 Fall 2012
Professor William Ott
Problem 1. (10) Let Sn = cfw_x Rn+1 : x = 1 and Dn+1 = cfw_x Rn+1 : x
1. Let X =
n
S [0, ). Show that one can dene an equivalence relation on X such that the quotient space
X
is homeomorphic to Dn+1 .
Prob
Assignment 2: Math 6342 Fall 2012
Professor William Ott
Problem 1. Let X and Y be topological spaces and let f : X Y be a map. Prove that if S is a subbasis for the
topology on Y such that f 1 (V ) is open for every V S, then f is continuous.
Problem 2. (
Assignment 3: Math 6342 Fall 2012
Professor William Ott
Problem 1. Let n > 1 be an integer. Show that Rn is not homeomorphic to R.
Problem 2. Let A R2 be a countable set. Show that R2 \ A is path connected.
Problem 3. Let X be a locally path connected top
Assignment 4: Math 6342 Fall 2012
Professor William Ott
Problem 1. Let X be a compact topological space. Let E = cfw_C : I be a collection of
nonempty closed subsets of X such that the intersection of any nite collection of elements of E is
itself a memb