2. Continuous Functions and Compact Sets Daisuke Oyama Mathematics - PowerPoint PPT Presentation

2. Continuous Functions and Compact Sets Daisuke Oyama Mathematics II April 8, 2020

Euclidean Norm in R N ▶ For x = ( x 1 , . . . , x N ) ∈ R N , the Euclidean norm of x is denoted by | x | or ∥ x ∥ , i.e., ( x 1 ) 2 + · · · + ( x N ) 2 , √ | x | = or ( x 1 ) 2 + · · · + ( x N ) 2 . √ ∥ x ∥ = ▶ We follow MWG to use ∥·∥ . ▶ For all x, y ∈ R N : ▶ ∥ x ∥ ≥ 0 ; ∥ x ∥ = 0 if and only if x = 0 ; ▶ ∥ αx ∥ = | α |∥ x ∥ for α ∈ R ; ▶ ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ . 1 / 45

Convergence in R N ▶ A sequence in R N is a function from N to R N . A sequence is denoted by { x m } ∞ m =1 , or simply { x m } , or x m . ▶ Notation (in this course): For A ⊂ R N , if x m ∈ A for all m ∈ N , then we write { x m } ∞ m =1 ⊂ A . Definition 2.1 x ∈ R N if A sequence { x m } ∞ m =1 converges to ¯ for any ε > 0 , there exists M ∈ N such that ∥ x m − ¯ x ∥ < ε for all m ≥ M. In this case, we write lim m →∞ x m = ¯ x or x m → ¯ x ( as m → ∞ ) . ▶ ¯ x is called the limit of { x m } ∞ m =1 . x ∈ R N is said to be convergent . ▶ A sequence that converges to some ¯ ▶ lim m →∞ x m = ¯ x if and only if lim m →∞ ∥ x m − ¯ x ∥ = 0 . 2 / 45

Convergence in R N Proposition 2.1 For a sequence { x m } in R N , where x m = ( x m 1 , . . . , x m N ) , x m → ¯ x N ) ∈ R N x = (¯ x 1 , . . . , ¯ if and only if x m i → ¯ x i ∈ R for all i = 1 , . . . , N . ▶ Thus, the definition in MWG (M.F.1) and that in Debreu (1.6.e) are equivalent. 3 / 45

Open Sets and Closed Sets in R N ▶ For x ∈ R N , the ε -open ball around x : B ε ( x ) = { y ∈ R N | ∥ y − x ∥ < ε } . Definition 2.2 ▶ A ⊂ R N is an open set if for any x ∈ A , there exists ε > 0 such that B ε ( x ) ⊂ A . ▶ A ⊂ R N is a closed set if R N \ A is an open set. Examples: ▶ { x ∈ R 2 | x 1 + x 2 < 1 } is an open set. { x ∈ R 2 | x 1 + x 2 ≤ 1 } is a closed set. ▶ B ε ( x ) , ε > 0 , is an open set. 4 / 45

Relative Openness and Closedness ▶ In Consumer Theory, for example, we usually work with R N + (set of nonnegative consumption bundles) rather than R N . ▶ We want to say { x ∈ R 2 | x 1 + x 2 < 1 , x 1 ≥ 0 , x 2 ≥ 0 } ( = { x ∈ R 2 + | x 1 + x 2 < 1 } ) is an open set in the world of R 2 + . Definition 2.3 For X ⊂ R N , ▶ A ⊂ X is an open set relative to X if for any x ∈ A , there exists ε > 0 such that ( B ε ( x ) ∩ X ) ⊂ A . ▶ A ⊂ X is a closed set relative to X if X \ A is an open set relative to X . 5 / 45

▶ Open sets, closed sets, and other concepts relative to X are defined with ▶ X in place of R N , and ▶ B ε ( x ) ∩ X in place of B ε ( x ) . ▶ A ⊂ X is an open set relative to X if and only if A = B ∩ X for some open set B ⊂ R N (relative to R N ). 6 / 45

Properties of Open Sets Proposition 2.2 Let X ⊂ R N . 1. ∅ and X are open sets relative to X . 2. For any index set Λ , if O λ is an open set relative to X for all λ ∈ Λ , then ∪ λ ∈ Λ O λ is an open set relative to X . (The union of any family of open sets is open.) 3. For any M ∈ N , if O m is an open set relative to X for all m = 1 , . . . , M , then ∩ M m =1 O m is an open set relative to X . (The intersection of any finite family of open sets is open.) 7 / 45

Properties of Closed Sets Proposition 2.3 Let X ⊂ R N . 1. ∅ and X are closed sets relative to X . 2. For any index set Λ , if C λ is a closed set relative to X for all λ ∈ Λ , then ∩ λ ∈ Λ C λ is a closed relative to X . (The intersection of any family of closed sets is closed.) 3. For any M ∈ N , if C m is a closed set relative to X for all m = 1 , . . . , M , then ∪ M m =1 C m is a closed set relative to X . (The union of any finite family of closed sets is closed.) 8 / 45

Properties of Closed Sets Proposition 2.4 Let X ⊂ R N . A ⊂ X is a closed set relative to X ⇒ for any convergent sequence { x m } ∞ ⇐ m =1 ⊂ A with x m → ¯ x ∈ X , we have ¯ x ∈ A . (A closed set is closed with respect to convergence.) 9 / 45

Proof ▶ By definition, A ⊂ X is a closed set relative to X ⇐ ⇒ ∀ x ∈ X \ A ∃ ε > 0 : B ε ( x ) ∩ A = ∅ . ▶ Therefore, if A is closed, then ∀ x ∈ X \ A , any sequence in A cannot converge to x . ▶ Conversely, if A is not closed, then ∃ ¯ x ∈ X \ A ∀ ε > 0 : B ε (¯ x ) ∩ A ̸ = ∅ . Then construct a sequence { x m } ∞ m =1 ⊂ A by x m ∈ B 1 m (¯ x ) ∩ A ( m = 1 , 2 , . . . ) . By construction, x m → ¯ ∈ A . x / 10 / 45

Interior, Closure, and Boundary Definition 2.4 For X ⊂ R N and A ⊂ X , ▶ the interior of A relative to X : Int X A = { x ∈ A | ( B ε ( x ) ∩ X ) ⊂ A for some ε > 0 } ; ▶ the closure of A relative to X : Cl X A = X \ Int X ( X \ A ) ; ▶ the boundary of A relative to X : Bdry X A = Cl X A \ Int X A . (We write Int R N = Int , Cl R N = Cl , and Bdry R N = Bdry .) 11 / 45

Characterization of Interior Proposition 2.5 Let X ⊂ R N and A ⊂ X . 1. Int X A ⊂ A . 2. Int X A is an open set relative to X . 3. If B ⊂ A and if B is open relative to X , then B ⊂ Int X A . Hence, ∪ Int X A = { B ⊂ X | B ⊂ A and B is open relative to X } , i.e., Int X A is the largest open set (relative to X ) contained in A . 12 / 45

Proof 1. By definition. ▶ Take any x ∈ Int X A . 2. By definition, ( B ε ( x ) ∩ X ) ⊂ A for some ε > 0 . We want to show that ( B ε ( x ) ∩ X ) ⊂ Int X A . ▶ Take any y ∈ B ε ( x ) ∩ X . Let ε ′ = ε − ∥ y − x ∥ > 0 . Then B ε ′ ( y ) ⊂ B ε ( x ) . ▶ Hence, ( B ε ′ ( y ) ∩ X ) ⊂ ( B ε ( x ) ∩ X ) ⊂ A , which implies that y ∈ Int X A . 3. Take any x ∈ B . By the openness of B , ( B ε ( x ) ∩ X ) ⊂ B for some ε > 0 . By B ⊂ A , ( B ε ( x ) ∩ X ) ⊂ A . Therefore, x ∈ Int X A . 13 / 45

Characterization of Closure Proposition 2.6 Let X ⊂ R N and A ⊂ X . 1. A ⊂ Cl X A . 2. Cl X A is a closed set relative to X . 3. If A ⊂ B and if B is closed relative to X , then Cl X A ⊂ B . Hence, ∩ Cl X A = { B ⊂ X | B ⊃ A and B is closed relative to X } , i.e., Cl X A is the smallest closed set (relative to X ) containing A . Proof ▶ By Proposition 2.5. 14 / 45

Examples ▶ For X = R , Int[0 , 1) = (0 , 1) , Cl[0 , 1) = [0 , 1] , Bdry[0 , 1) = { 0 , 1 } . ▶ What are the interior, closure, and boundary of Q ∩ [0 , 1] ? → Homework ▶ For A = { ( x 1 , x 2 ) ∈ R 2 | 0 ≤ x 1 ≤ 1 , x 2 = 0 } , Int A (= Int R 2 A ) = ∅ , while Int R A = (0 , 1) . Remark There is an abuse of notation in “ Int R A = (0 , 1) ”: To be precise, one should write Int { x ∈ R 2 | x 1 ∈ R , x 2 =0 } A = { x ∈ R 2 | x 1 ∈ (0 , 1) , x 2 = 0 } . 15 / 45

Characterizations of Open/Closed Sets by Interior/Closure Proposition 2.7 Let X ⊂ R N and A ⊂ X . 1. A is open relative to X ⇐ ⇒ Int X A = A . 2. A is closed relative to X ⇐ ⇒ Cl X A = A . 16 / 45

Characterizations of Closure Proposition 2.8 Let X ⊂ R N and A ⊂ X . 1. Cl X A = { x ∈ X | B ε ( x ) ∩ A ̸ = ∅ for all ε > 0 } ∩ B ε ( A ) ∩ X , = ε> 0 where B ε ( A ) = { x ∈ R N | ∥ x − a ∥ < ε for some a ∈ A } . 2. Cl X A = { x ∈ X | x m → x for some { x m } ⊂ A } . ▶ Thus, the definition in MWG and that in Debreu are equivalent. 17 / 45

Proof of Proposition 2.8 Recall the definition: Cl X A = X \ Int X ( X \ A ) . 1. x ∈ Cl X A ⇐ ⇒ x / ∈ Int X ( X \ A ) ⇐ ⇒ ∀ ε > 0 : ( B ε ( x ) ∩ X ) ̸⊂ ( X \ A ) ⇐ ⇒ ∀ ε > 0 : B ε ( x ) ∩ A ̸ = ∅ ⇐ ⇒ ∀ ε > 0 : x ∈ B ε ( A ) . 2. If x ∈ Cl X A , construct { x m } ⊂ A by x m ∈ B 1 m ( x ) ∩ A , m ( x ) ∩ A ̸ = ∅ by part 1. where B 1 Then x m → x . Conversely, let { x m } ⊂ A and x m → x . For any ε > 0 , there exists M such that x M ∈ B ε ( x ) , so that B ε ( x ) ∩ A ̸ = ∅ . Hence, x ∈ Cl X A by part 1. 18 / 45

Dense Sets Definition 2.5 For X ⊂ R N , A ⊂ X is dense in X if Cl X A = X . Proposition 2.9 For X ⊂ R N and A ⊂ X , the following statements are equivalent: 1. A is dense in X . 2. Int X ( X \ A ) = ∅ . 3. O ∩ A ̸ = ∅ for every nonempty open set O ⊂ X relative to X . Proof ▶ By the definitions of interior and closure. 19 / 45

Compact Sets ▶ A ⊂ R N is bounded if there exists r ∈ R such that ∥ x ∥ < r for all x ∈ A . Definition 2.6 A ⊂ R N is compact if it is bounded and closed (relative to R N ). Examples: ▶ [0 , 1] ⊂ R is compact. ▶ [0 , ∞ ) ⊂ R is not compact. ▶ (0 , 1] ⊂ R is not compact. 20 / 45

Sequential Compactness Proposition 2.10 For A ⊂ R N , the following statements are equivalent: 1. A is compact. 2. For every sequence { x m } ⊂ A , there exist a subsequence { x m ( k ) } of { x m } and a point x ∈ A such that x m ( k ) → x . 21 / 45

Proof 2 ⇒ 1 If A is not bounded, then for all m ∈ N , there exists x m ∈ A such that ∥ x m ∥ ≥ m . No subsequence of the sequence { x m } ⊂ A can be convergent ( ∵ ∀ x ∈ R N ∃ M ∈ N : ∥ x m − x ∥ ≥ ∥ x m ∥ − ∥ x ∥ ≥ 1 ∀ m ≥ M ) . If A is not closed, then there exists ¯ x / ∈ A such that for all m ∈ N , there exists x m ∈ B 1 x ) ∩ A . m (¯ The sequence { x m } ⊂ A , and any subsequence, converges to ¯ x / ∈ A . 22 / 45