Lecture-12: Closed Sets

Transcription

1 and Its Examples Properties of Lecture-12: Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014

2 Outline Introduction and Its Examples Properties of 1 Introduction 2 and Its Examples 3 Properties of 4

3 and Its Examples Properties of As we studies lots of thing about open sets now we are going to discuss about complementary concept and introduce closed sets.

4 and Its Examples Properties of and Its Examples I Definition A subset A of a topological space X is closed if the set X A is open.

5 and Its Examples Properties of 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.

6 and Its Examples Properties of and Its Examples III Example Closed balls Go and closed rectangles Go are closed sets in the standard topology on R 2. Let A = [a, b] [c, d] be a closed rectangle in R 2. To show that A is a closed set in the standard topology, we prove that R 2 A is an open set. Note that R 2 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 R 2 A is an open set. Hence, the rectangle A is a closed set.

7 Introduction and Its Examples Properties of and Its Examples IV Which are closed and which are open?

8 and Its Examples Properties of and Its Examples V The spiral is the graph of the polar-coordinates equation r = θ θ + 1, θ 0 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.

9 and Its Examples Properties of 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?

10 Introduction and Its Examples Properties of 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.

11 and Its Examples Properties of 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.

12 and Its Examples Properties of Properties of I Since closed sets are complementary to open sets, their properties are similar, but there are some fundamental differences.

13 and Its Examples Properties of Properties of 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 F i be subset of X for all i in N and G = i=1 F i and H = n i=1 F i. Now F i is closed in X (X F i ) is open n (X F i ) and (X F i ) are open sets (by Theorem) i=1 i=1 n X F i and X F i are open sets (by De Morgan s Law) i=1 i=1 n F i and F i are closed sets i=1 i=1 Hence G and H are closed sets.

14 and Its Examples Properties of Properties of 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 F n = [0, n+1 ] for all n in N be subset of R. Here F n is closed. And F i = i=1 [ 0, 1 ] [ 0, 2 ]... [0, 1) 2 3 = [0, 1) closed set Therefore, i=1 F i is not closed set, even each F n is a closed set.

15 and Its Examples Properties of Definition (Closed Ball) For each x in R 2 and ɛ > 0, define the closed ball of radius ɛ centered at x to be the set B(x, ɛ) = {y R 2 : d(x, y) < ɛ} 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] R 2 is called a closed rectangle.