Ans-10 - Answers Assignment#10 Exercises 1.4 1(a True If s...

Answers, Assignment #10
Exercises 1.4
1. (a) True. If s ∈ int S , then s does not belong to bd S .
(b) False. Let S = (0 , 1). Then bd S = { 0 , 1 } which is not a subset of S .
(c) False. S = [0 , 1], bd S = { 0 , 1 } , S = bd S .
(d) True. Suppose s ∈ S . If s / ∈ int S , then every neighborhood of s must intersect S c
which implies s ∈ bd S .
(e) True. s ∈ bd S implies N ∩ S = ∅ and N ∩ S
c
= ∅ for every neighborhood N of s .
t ∈ bd S c implies N ∩ S c = ∅ and N ∩ ( S c ) c = N ∩ S = ∅ for every neighborhood N
of t . Therefore bd S = bd S
c
.
(f) False. S = [0 , 1], bd S = { 0 , 1 } ⊂ S .
2. (a) True. Let z ∈ N ( x, ). Let δ = min {| z - x | , | z - ( x + ) | , | z - ( x - ) |} . Then
N ( z, δ ) ⊂ N ( x, ).
(b) True. Theorem 10 (a).
(c) False.
∞
n =1
1
n
, 1 -
1
n
= (0 , 1).
(d) False.
∞
n =1
(
-
1
n
, 1 +
1
n
)
= [0 , 1].
(e) True. Let T be a collection of closed sets. Then [ ∩ T
T ∈T
]
c
= ∪ T
c
t ∈T
. T closed implies
T c is open and the union of a collection of open sets is open. Therefore, [ ∩ T
T ∈T
]
c
is
open and ∩ T
T ∈T
is closed.
(f) False. The set of real numbers is both open and closed.