Masters Comprehensive Exam - Topology
January, 2010
CHOOSE THREE
In the following, R n denotes n-dimensional Euclidean space with the usual topology.
(1) For each n ∈ N let X n := {(1/n, y) : 0 ≤ y ≤ 1}, let X∞ := {(0, y) : 0 ≤ y ≤ 1},
and let Y0 := {(x, 0) : 0 ≤ x ≤ 1}. Define C to be the subspace of R2 given by
C := X∞ ∪ (∪ n∈N X n) ∪ Y0 . Prove that there is no continuous map from the unit interval
[0, 1] onto C.
(2) Let ∼ denote the equivalence relation on R n \ {o} defined by x ∼ y iff there is a t ∈
such that x = ty. Let P n be the quotient space P n := (R n \ {o})/ ∼.
(a) Prove that P n is connected.
(b) Prove that P n is compact.
R
(3) A continuous map α : [a, b] → R n is a piece-wise linear arc from x to y provided there
is an n ∈ N, there are t i in [a, b] with a = t 0 < t 1 < · · · < t n = b, and xi ∈ R n, such
that x0 = x, x n = y, and
α(t) = (1 −
(t − t i )
(t i+1 − t i )
and i = 0, . . . , n − 1.
)xi +
(t − t i )
(t i+1 − t i )
xi+1 ,
for t i ≤ t ≤ t i+1
Prove that if U is a connected open set in R n, then U is piece-wise linearly arcconnected (that is, for each x and y in U, there is a piece-wise linear arc from x to
y with image in U).
(4) (TRUE or FALSE) Decide for each statement below if it is true or false and support your
conjecture by either a proof or a counterexample.
a.: Each contraction is uniformly continuous. (A map f : X → X, where (X, d) is a
metric space is a contraction if there is α < 1 such that d( f (x), f (y)) ≤ αd(x, y)
for all x, y ∈ X with x , y.)
b.: Every closed set is compact.
c.: If g : X → Y is an inclusion, then π1 (X, x0 ) → π1 (Y, x0 ) is an injection.
d.: If Y is contractible, then for all topological spaces X, [X, Y] has a single element.
( Y is said to be contractible if the identity map iY : Y → Y is null-homotopic.)