Download §2.1–Unions and Intersections Of Open and Closed Sets

Open sets
Closed sets
§2.1–Unions and Intersections
Of Open and Closed Sets
Tom Lewis
Fall Semester
2014
Open sets
Closed sets
Outline
Open sets
Closed sets
Open sets
Closed sets
Example
Let {qi , i ∈ N} be a listing of the rational numbers in [0, 1]. Let
Ai = (qi − 1/4i , qi + 1/4i ) and let
A=
∞
[
Ai .
i=1
Is A open? Does A contain [0, 1]?
Open sets
Closed sets
Theorem
An arbitrary (finite, countable, or uncountable) union of open sets
is open.
Open sets
Closed sets
Problem
Construct an example to show that an infinite intersection of open
sets need not be open.
Open sets
Theorem
The intersection of a finite number of open sets is open.
Closed sets
Open sets
Closed sets
Theorem
An arbitrary (finite, countable, or uncountable) intersection of
closed sets is closed.
Open sets
Theorem
The union of a finite number of closed sets is closed.
Closed sets
Open sets
Closed sets
Example (Cantor set)
Let C0 = [0, 1], C1 = [0, 1/3] ∪ [2/3, 1], and
C2 = [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [8/9, 1]
|
{z
} |
{z
}
1
3 C1
1
2
3 C1 + 3
In general, let
Cn =
1
Cn−1
3
∪
1
2
Cn−1 +
3
3
.
Let C = ∩n>1 Cn , called Cantor’s middle-third set. Show that C is
a closed set.