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Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Lecture-12: Closed Sets
Dr. Sanjay Mishra
Department of Mathematics
Lovely Professional University
Punjab, India
October 18, 2014
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Outline
1
Introduction
2
Closed Sets and Its Examples
3
Properties of Closed Sets
4
Appendix
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
As we studies lots of thing about open sets now we are going to discuss
about complementary concept and introduce closed sets.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples I
Definition
A subset A of a topological space X is closed if the set X − A is open.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples II
Example
Consider R with the standard
topology. Then,
1
Since (0, 1) is open,
(−∞, 0] ∩ [1, (0) is closed.
2
Since (−∞, a) ∪ (b, ∞) is open,
[a, b] is closed.
3
Since (−∞, c) ∪ (c, ∞) is open,
{c} is closed.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples III
Example
Closed balls Go and closed rectangles
standard topology on R2 .
Go
are closed sets in the
Let A = [a, b] × [c, d] be a closed rectangle in R2 . To show that A is a
closed set in the standard topology, we prove that R2 − A is an open set.
Note that R2 − A can be expressed as the union of four open half-planes:
{(x, y) : x < a}, {(x, y) : x > b}, {(x, y) : y < c}, and {(x, y) : y > d}
Since each of these half-planes is an open set and a union of open sets is
an open set, it follows that R2 − A is an open set. Hence, the rectangle
A is a closed set.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples IV
Which are closed and which are
open?
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples V
The spiral is the graph of the polar-coordinates equation
r=
θ
,θ ≥ 0
θ+1
It winds out toward the circle S 1 , but it contains no point from S 1 . The
spiral is not closed because the point (1, 0) is in the complement, but
no open ball containing (1, 0) is a subset of the complement since every
such open ball intersects the spiral. Thus the complement is not open,
implying that the spiral is not closed.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples VI
If a set C is closed, then by definition its complement is open.
What can we say about the complement of an open set?
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples VII
The complement of a closed set is
open by the definition of closed set,
and the complement of an open set
is closed by the argument just
presented.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Closed Sets and Its Examples VIII
Let the set X = {a, b, c, d} have the topology
τ = {φ, X, {b}, {a, b}, {b, c, d}, {c, d}}
Note that {b} is open and not closed, {a} is closed and not open, {a, b}
is both open and closed, and {b, c} is neither open nor closed.
Question: How is a subset different from a door?
Answer: A door must be open or closed. But a subset can be either,
both, or neither.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Properties of Closed Sets I
Since closed sets are complementary to open sets, their properties are
similar, but there are some fundamental differences.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Properties of Closed Sets II
Theorem
In a topological space (X, τ ), an arbitrary intersection of closed sets
and a finite union of closed sets is open.
Proof:
Let Fi be subset of X for all i in N and G =
T∞
i=1
Fi and H =
Sn
i=1
Fi . Now
Fi is closed in X ⇒ (X − Fi ) is open
⇒
∞
[
(X − Fi ) and
i=1
n
\
⇒X−
∞
\
Fi and X −
i=1
⇒
∞
\
i=1
(X − Fi ) are open sets
(by Theorem)
i=1
Fi and
n
[
Fi are open sets
i=1
n
[
Fi are closed sets
i=1
Hence G and H are closed sets.
Sanjay Mishra
Closed Sets
(by De Morgan’s Law)
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Properties of Closed Sets III
Theorem
The union of infinite collection of closed sets in a topological space is
not necessarily closed.
Proof:
n
Let (R, τ ) be topological space and Fn = [0, n+1
] for all n in N be subset
of R. Here Fn is closed. And
∞
[
i=1
1
2
Fi = 0,
∪ 0,
∪ . . . ∪ [0, 1)
2
3
= [0, 1) 6= closed set
Therefore,
S∞
i=1 Fi
is not closed set, even each Fn is a closed set.
Sanjay Mishra
Closed Sets
Introduction
Closed Sets and Its Examples
Properties of Closed Sets
Appendix
Definition (Closed Ball)
For each x in R2 and > 0, define the closed ball of radius centered
at x to be the set
B̄(x, ) = {y ∈ R2 : d(x, y) < }
where d(x, y) is the Euclidean distance between x and y.
Definition (Closed Rectangle)
If [a, b] and [c, d] are closed bounded intervals in R, then the product
[a, b] × [c, d] ⊂ R2 is called a closed rectangle.
Sanjay Mishra
Closed Sets