Math 434/542
Fall 2015
Test 2, 11-16-15
Carter
1. Recall that the usual topology on Rn has as a basis the set of open balls


v
u n


uX
(yi − xi )2 < B (~x) = ~y ∈ Rn : t


i=1
where > 0 and ~x = (x1 , x2 , . . . , xn ) ranges over all points in Rn .
In R2 consider the set
(
Dδ (~x) =
~y ∈ R2 :
n
X
)
|yi − xi | < δ
.
i=1
(a) Show Dδ (~x) is open in the usual topology for R2 .
(b) Let D denote the topology with basis
{Dδ (~x) : δ > 0 & ~x ∈ R2 }.
Show B (~x) is open in D.
2. Consider
S = {(x, y) ∈ R2 : x2 + y 2 < 1 & y 6= 0}
as a subset of R2 in the usual topology.
(a) Is the set connected? Prove your assertion.
(b) Is R2 \ S connected?
(c) Is S open in R2 ? Prove your assertion.
(d) Describe the closure of S.
3. Consider C to be the Cantor set as a subset of R, and consider Q ∩ [0, 1] — the set of
rational numbers in the unit interval.
(a) Is C closed?
(b) Is Q ∩ [0, 1] closed?
(c) Is Q ∩ [0, 1] discrete?
(d) Is C discrete?
(e) Are either C or Q ∩ [0, 1] connected?
(f) Are C and Q ∩ [0, 1] homeomorphic?
Justify your answers.
1