Igor Zelenko, Fall 2009
1
Homework Assignment 1 in Topology I, MATH636
due to Sept 7, 2009
1.
a. Find an example of a metric space (X, d) containing two points x1 6= x2 such that
for some real r > 0 the ball of radius r with center at x1 coincides with the ball
of radius r with center at x2 , i.e. Br (x1 ) = Br (x2 );
b. Prove or disprove by giving a counter-example: If d1 and d2 are metrics on a set
X, then the function d(x, y) = min{d1 (x, y), d2 (x, y)} defines a metric on X.
c. Show that if (X, d) is a metric space, then ρ(x, y) =
d(x,y)
1+d(x,y)
is a metric in X and
ρ and d are equivalent metric, i.e. they induce the same topology on X.
2.
a. Let X be any set, and define Tf = {U ⊂ X : X\U is finite} ∪ {∅}. Show that Tf
is a topology on X.
b. Is the collection T∞ = {U ⊂ X : X\U is infinite} ∪ {X, ∅} a topology on X? Give
a proof to your answer.
3. A map f : X → Y between two topological spaces is open if whenever U ⊂ X is open,
then f (U ) ⊂ Y is open in Y . Consider X = R2 with the lexicographic (dictionary)
order (cf. p. 6, example 6 in the text) and R with its usual topology. Let f : X → R
and g : X → R be the projection onto the first and second factors, respectively. (That
is, f (x, y) = x and g(x, y) = y.) Show that f is not an open map and g is an open
map.
4. Let D denote the unit ball B1 (0) centered at 0 ∈ R2 with the Euclidean metric.
a. Show that D is homeomorphic to R2 .
b. Show that D is homeomorphic to the upper half plane H = {(x, y) ∈ R2 : y > 0}.