Walheer Barnabé. Topics in Mathematics Practical Session 2 - Topology & Convex

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1 Topics in Mathematics Practical Session 2 - Topology & Convex Sets

2 Outline (i) Set membership and set operations (ii) Closed and open balls/sets (iii) Points (iv) Sets (v) Convex Sets

3 Set Membership and Set Operations (i) x S indicates that x is an element of S and x S indicates that x is not an element of S. (ii) A B indicates that A is a subset of B. (iii) A B indicates the union of A and B i.e. the elements that belongs to at least one of the sets A and B. (iv) A B indicates the intersection of A and B i.e. the elements that belongs to both A and B. (v) A \ B indicates A minus B i.e. the elements that belongs to A, but not to B.

4 Exercises Give a explicit definition of (ii), (iii), (iv) and (v). Give the definition of a proper subset (A B). Prove that: A = B iff A B and B A. Prove that A A.

5 Closed and open balls/sets B(c, r) = {x R n : d(c, x) < r} is an open ball of R n. B(c, r) = {x R n : d(c, x) r} is a closed ball of R n. (r > 0) A is closed iff all sequences of elements of A that converge, converge in A. B is open iff its complement (B c ) is closed [B c = R n \ B]. An open (closed) ball is a open (closed) set. If A and B are open (closed), then: A B is open (closed). A B is open (closed).

6 Points a is an interior point of A iff a is the center of an open ball in A. Int A = set of all interior points of A. This implies: (i) If a A, then a cannot be an interior point of A. (ii) All points of an open set A are interior points of A. (iii) If all points of A are interior points, then A is open. (iv) The set A is open iff all these points are interior points.

7 Points b is an adherent (or closure) point of B iff there exists a sequence of elements of B that converge through b. Adh B = set of all adherent points of B. Or use: b is an adherent point of B iff all interval centered in b has a non-empty intersection with B. This implies: (i) If b B, then b is an adherent point of B. (ii) The set B is closed iff it contents all these adherent points.

8 Duality: Interior vs Adherent The interior of S is the complement of the set of all adherent points of the complement of S. In this sense Int and Adh are dual notions (kind of involution).

9 Points c is an accumulation (or limit) point of C iff there exists a sequence of elements of C, different of c, that converge through c. Acc C = set of all accumulation points of C. Or use: c is an accumulation point of C iff all interval centered in c has a non-empty intersection with C that is not c. This implies: (i) If c is an acccumulation point of C, then c is an adherent point of C.

10 Exercises: Find Int, Adh and Acc of the following sets. Are the sets open, closed? In R: A = [2, 5] [7, 12], B = {1}, C = ( 1, 3) (0, 2), D = [ 4, 1] {1, 2}, E = R \ [3, 7], F = (, 4] [6, + ) In R 2 : A = R 0 x R, B = (R + ) 2, C = {(x, y) R 2 : 2x + 3y 6, x 0, y 0}, D = {(x, y) R 2 : 0 x 3, 0 y 2} {(3, 3)}, E = {(x, y) R 2 : 1 < x 2 + y 2 < 2} In R 3 : A = (R + 0 )3, B = {(x, y, z) R 3 : 0 x 3, 0 y 2, 0 z 1}, C = {(x, y, z) R 3 : z = x 2 + y 2 }

11 Min, Max, Sup and Inf M is a maximum of A iff M A a A : a M. m is a minimum of A iff m A a A : a m. s is a supremum of A iff a A : a s x R n, a A : x a x s. i is a infimum of A iff a A : a i x R n, a A : x a x i. A is a complete set iff it has an infimum and a supremum.

12 Exercises: Find Min, Max, Sup and Inf of the following sets. A = [0, 1], B = ( 1, 1), C = [ 2, 69), D = R, E = N, F = N 0 Show that sup is unique (using the definition). Derive the relation between inf and min (using the definitions). Is Q complete?

13 Sets A set is bounded iff it is included in a ball. A set is compact iff it is closed and bounded. The neighborhood of a is the set that containts an open ball centered in a. The frontier (or boundary) of A (Fr A or Bd A) is Adh A \ Int A. The exterior of a set is the interior of its complement. The power set of S is the set of all subsets of S (P(S)) 2 n elements.

14 Exercises: Find Fr of the following sets. Are the sets bounded? In R: A = [2, 5] [7, 12], B = {1}, C = ( 1, 3) (0, 2), D = [ 4, 1] {1, 2}, E = R \ [3, 7], F = (, 4] [6, + ) In R 2 : A = R 0 x R, B = (R + ) 2, C = {(x, y) R 2 : 2x + 3y 6, x 0, y 0}, D = {(x, y) R 2 : 0 x 3, 0 y 2} {(3, 3)}, E = {(x, y) R 2 : 1 < x 2 + y 2 < 2} In R 3 : A = (R + 0 )3, B = {(x, y, z) R 3 : 0 x 3, 0 y 2, 0 z 1}, C = {(x, y, z) R 3 : z = x 2 + y 2 }

15 Convex Set A is convex iff x, y A, λ [0, 1], λx + (1 λ)y A. This implies: (i) The line segment passing through two points of A is in A. If A and B are convex, then A B is convex. A B is not convex. V is a cone iff x V, λ R, λx V. A convex hull of W is the smallest convex set that contains W.

16 Exercises: Are the following sets convex? In R: A = [2, 5] [7, 12], B = {1}, C = ( 1, 3) (0, 2), D = [ 4, 1] {1, 2}, E = R \ [3, 7], F = (, 4] [6, + ) In R 2 : A = R 0 x R, B = (R + ) 2, C = {(x, y) R 2 : 2x + 3y 6, x 0, y 0}, D = {(x, y) R 2 : 0 x 3, 0 y 2} {(3, 3)}, E = {(x, y) R 2 : 1 < x 2 + y 2 < 2} In R 3 : A = (R + 0 )3, B = {(x, y, z) R 3 : 0 x 3, 0 y 2, 0 z 1}, C = {(x, y, z) R 3 : z = x 2 + y 2 }

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