1. hw1 Deadline: 9/21/5:00pm
Let x = (x1, · · · , xn) and y = (y1, · · · , yn) be two points of Rn. The Euclidean distance between x and y is defined by
d(x, y) =
n
X
i=1
(xi− yi)2
!1
2
.
Let be a positive number. The Euclidean open ball center at x of radius is the set of points y ∈ Rn so that d(x, y) < , i.e.
B(x, ) = {y ∈ Rn: d(x, y) < }.
Let A be a subset of Rnand x ∈ A be a point. Recall that x is an interior point of A if there exists > 0 so that B(x, ) ⊂ A. The set of all interior points of A is denoted by int(A). A subset U of Rn is open if U = int(U ).
(1) Sketch the given subset A of R2 and prove that A is open.
(a) A = {(x, y) ∈ R2: ax + by > c}. Here a, b, c are real numbers so that a2+ b2 6= 0.
(b) A = {(x, y) ∈ R2: xy > 1}.
(c) A = {(x, y) ∈ R2: r2 < x2+ y2 < R2}. Here R > r > 0.
(2) Let A, B be subset of Rn. We define
A + B = {a + b ∈ Rn: a ∈ A, b ∈ B}.
When A = {a}, we simply denote A + B by a + B.
(a) Let U be an open subset of Rn. Show that x + U is open for any x ∈ Rn. (b) Show that A + U is open for any open subset U of Rn.
(3) Let C be the subset of R2 consisting of points (x, y) such for each (x, y) ∈ C, there exists k ∈ N with (x, y) ∈ B((1/k, 0), 1) centered at (1/k, 0) of radius 1 i.e.
C = {(x, y) ∈ R2 : there exists k ∈ N such that (x, y) ∈ B((1k, 0), 1)}.
Show that C is open.
(4) Let ai, bi, ci, 1 ≤ i ≤ m be real numbers such that a2i + b2i 6= 0 for 1 ≤ i ≤ m. Show that
A = {(x, y) ∈ R2 : aix + biy < ci, 1 ≤ i ≤ m}
is open. Hint: do the case when A = {(x, y) ∈ R2 : 2x + 5y < 1, x − y < 2}
(5) Let k be any natural numbers. Set Bk=
(x, y) ∈ R2 : x2+ y2 < 1 k
, k ∈ N.
(a) Show that
m
\
k=1
Bk is open for any m ≥ 1.
(b) Find
∞
\
k=1
Bk and show that
∞
\
k=1
Bk is not open.
Remark. Cite Definitions/Lemmas/Propositions/Theorems proved in class as many as possible; You need to indicate which Definitions/Lemmas/Propositions/Theorems you are using.
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