Shawn Zhong

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Shawn Zhong

钟万祥
  • Tutorials
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Math 521

Home / Mathematics / Notes / Math 521 / Page 6

Math 521 - 2/26

  • Feb 27, 2018
  • Shawn
  • Math 521
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Theorem 2.24 (a) For any collection {G_n } of open sets, ⋃8_α▒G_α is open ○ Suppose G_α is open for all α ○ Let G=⋃8_α▒G_α ○ If x∈G, then x∈G_α for some α ○ Since G_α is open, there is a neighborhood about x in G_α ○ And consequently, the neighborhood about x is also in G ○ Thus G is open (b) For any collection {F_n } of closed sets, ⋂8_α▒F_α is closed ○ Suppose F_α is closed for all α ○ Then F_α^c is open by Theorem 2.23 ○ So ⋃8_α▒F_α^c is open by (a) ○ (⋂8_α▒F_α )^c=⋃8_α▒F_α^c , by De Morgan^′ s Law ○ Thus, (⋂8_α▒F_α )^c is open ○ Therefore ⋂8_α▒F_α is closed by Theorem 2.23 (c) For any finite collection, G_1,G_2,…,G_n of open sets, ⋂8_(i=1)^n▒G_i is also open ○ Suppose G_1,G_2,…,G_n is open ○ Let x∈H=⋂8_(i=1)^n▒G_i ○ So, x∈G_i for 1≤i≤n ○ By definition, since each G_i is open ○ x is contained in a neighborhood N_(r_i ) (x)⊂G_i ○ Let r=min⁡(r_1,r_2,…,r_n ) ○ N_r (x)⊂G_i for 1≤i≤n ○ So, N_r (x)∈H ○ Thus, H=⋂8_(i=1)^n▒G_i is open (d) For any finite collection, F_1,F_2,…,F_n of closed sets, ⋃24_(i=1)^n▒F_i is also closed ○ Suppose F_1,F_2,…,F_n is closed ○ Then F_i^c is open by Theorem 2.23 ○ So ⋂24_(i=1)^n▒F_i^c is open by (c) ○ (⋃24_(i=1)^n▒F_i )^c=⋂24_(i=1)^n▒F_i^c , by De Morgan^′ s Law ○ Thus, (⋃24_(i=1)^n▒F_i )^c is open ○ Therefore ⋃24_(i=1)^n▒F_i is closed by Theorem 2.23 • Note ○ ⋂24_(n=1)^∞▒(−1/n,1/n) ={0} ○ (−1/n,1/n) is open ∀n∈N, while {0} is closed Closure • Let X be a metric space • If E⊂X and E′ denotes the set of limit points of E in X • Then the closure of E is defined to be E ̅=E∪E^′ Theorem 2.27 • If X is a metric space and E⊂X, then • E ̅ is closed ○ Let p∈E ̅^c ○ Then p is neither a point of E nor a limit point of E ○ So there exists a neighborhood N about p that contains no points of E ○ So,N⊂E ̅^c ○ i.e. every point of E ̅^c is an interior point ○ Thus E ̅^c is open ○ Therefore E ̅ is closed • E=E ̅ iff E is closed ○ If E=E ̅, then E is closed ○ If E is closed, E contains its limit points, so E^′⊂E and E=E ̅ • E ̅⊂F for every closed set F⊂X s.t. E⊂F ○ Suppose F is closed and E⊂F ○ F is closed⇒F^′⊂F ○ E⊂F⇒E^′⊂F′⊂F ○ Thus E ̅=E∪E^′⊂F • Intuition: E ̅ is the smallest closed set in X containing E Theorem 2.28 • Statement ○ If E≠∅, E⊂R, and E is bouned above, then sup⁡E∈E ̅ ○ Hence sup⁡E∈E if E is closed • Proof ○ Let y=sup⁡E ○ If y∈E § Clearly y∈E ̅ ○ If y∉E § Let h 0 § Let x∈(y−hy) § Suppose ∄x∈E, then y−h is an upper bound for E § But this contradicts the fact that y=sup⁡E § So there must be some x∈E with y−h x y § Thus, for any neighborhood about y, ∃x∈E in the neighborhood § So y is a limit point of E § i.e. y∈E^′⊂E ̅
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Math 521 - 2/23

  • Feb 23, 2018
  • Shawn
  • Math 521
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Theorem 2.19 • Statement ○ Every neighborhood is an open set • Proof ○ Let X be a metric space ○ Choose neighborhood N_r (p)=E∈X ○ Let q∈E ○ Choose h s.t. d(p,q)=r−h ○ Consider N_h(q) ○ So, if s∈N_h(q), d(q,s)h ○ d(p,s)≤d(p,q)+d(q,s)r−h+h=r ○ Thus d(p,s)r ○ i.e. s∈N_r (p) ○ So N_h(q)⊂N_r (p) ○ Therefore N_r (p) is open Theorem 2.20 • Statement ○ If p is a limit point of E, then ○ every neighborhood of p contains infinitely many points of E • Proof ○ Suppose the opposite ○ Then there exists a set E with a limit point p s.t. ○ The neighborhood of p contains only finitly many points of E ○ Namely q_1,q_2,…,q_n ○ Let r=min⁡(d(p,q_1 ),d(p,q_2 ),…,d(p,q_n )) ○ By definition, q_i∉N_r (p) for 1≤i≤n ○ This contradicts the fact that p is a limit point ○ So, this neighborhood about p must contain infinitely many points • Corollary ○ A finite set has no limit points Theorem 2.22 (De Morgan s Law) • Statement ○ Let {E_x } be a finite or infinite collection of sets, then ○ (⋃8_α▒E_α )^c=⋂8_α▒(E_α )^c • Proof ○ Suppose x∈(⋃8_α▒E_α )^c § Then x∉⋃8_α▒E_α § So x∉E_α for all α § Thus, x∈(E_α )^c for all α § So, x∈⋂8_α▒(E_α )^c § Therefore (⋃8_α▒E_α )^c⊂⋂8_α▒(E_α )^c ○ Suppose x∈⋂8_α▒(E_α )^c § Then x∈(E_α )^c for all α § So x∉E_α for all α § x∉⋃8_α▒E_α § Thus, x∈(⋃8_α▒E_α )^c § So ⋂8_α▒(E_α )^c ⊂(⋃8_α▒E_α )^c ○ Therefore (⋃8_α▒E_α )^c=⋂8_α▒(E_α )^c Theorem 2.23 • Statement ○ A set E is open if and only if E^c is closed ○ Note: This does not say that open is not closed and closed is not open • Proof ○ Suppose E^c is closed § Choose x∈E, so x∉E^c § So, x is not a limit point of E^c § So, there exists a neighborhood N_r (x) that contains no points of E^c § So, N_r (x)∩E^c=∅ § Consequently, N_r (x)⊂E § So, x is an interior point of E § By definition, E is open ○ Suppose E is open § Let x be a limit point of E^c (if exists) § So, every neighborhood of x contains a point in E^c § So, x is not an interior point of E § E is open, so x∈E^c § Thus, E^c contains its limit points and is closed by definition • Corollary ○ A set E is closed if and only if E^c is open Examples 2.21 • Let X=R2 Subset Closed Open Perfect Bounded {x ⃗∈R2│|x ⃗ |1} × ✓ × ✓ {x ⃗∈R2│|x ⃗ |≤1} ✓ × ✓ ✓ A nonempty finite set ✓ × × ✓ Z ✓ × × × {1/n│n∈N × × × ✓ R2 ✓ ✓ ✓ × (a,b) × ? × ✓ • Note: (a,b) is open as a subset of R, but not as a subtset of R2
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Math 521 - 2/21

  • Feb 21, 2018
  • Shawn
  • Math 521
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Definitions 2.18 • Let X be a metric space. All points/elements below are in X • Neighborhood ○ Definition § A neighborhood of p is a set N_r (p) § consisting of all points q such that d(p,q)r for some r∈R § r is the radius of N_r (p) ○ Example: R2 ○ Example: Taxicab metric • Limit point ○ Definition § A point p is a limit point of the set E⊂X if § every neighborhood of p contains a point q∈E and p≠q ○ Example: R2 ○ Example: (0,1)∈R § For (0,1)∈R, the limit points is [0,1] • Isolated point ○ Definition § If p∈E and p is not a limit point of E § then p is an isolated point of E ○ Example: Z in R § Every integers is an isolated point in R • Closed set ○ Definition § A set E is closed if every limit point of E is in E ○ Example: [0,1]∈R § In R, neighborhood of p∈R are open intevals cenerted about p § All of [0,1] is a limit point since § If x∈[0,1] □ The neighborhood about x is (x−r,x+r) □ (x−r,x+r)∩[0,1] is non-empty □ If x=0, then take q=min⁡(x+r/2,1) □ Otherwise take q=max⁡(x−r/2,0) □ So every point in [0,1] is a limit point § If x∉[0,1] □ i.e. x0 or x1 □ Take r={■8(|x|&if x0@|x−1|&if x1)┤ □ Then N_r (x)∩[0,1]=∅ □ So nothing outside of [0,1] is a limit point of [0,1] § So [0,1] contains all its limit points § Thus [0,1] is closed • Interior point ○ Definition § A point p is an interior point of a set E if § there exists a neighborhood N_r (p) that is a subset of E ○ Example: R2 § For the closed set S § The point x is an interior point of S § The point y is not an interior point of S (on the boundary of S) • Open set ○ Definition § E is an open set if every point of E is an interior point ○ Example: R2 § U is an open set, since ∀x∈U, ∃B_ϵ (x)⊂U ○ Example: (0,1)∈R § For x∈(0,1) § Take r=min⁡(x,1−x) § N_r (x)⊂(0,1) § Thus every point in (0,1) is an interior point • Complement ○ Definition § The complement of E (denoted E^c) is {p∈X│p∉E} • Perfect ○ Definition § E is perfect if E is closed and every point of E is limit point of E • Bounded ○ Definition § E is bounded if there is a real number M and a point p∈E s.t. § d(p,q)M for all p∈E • Dense ○ Definition § E us dense in X if every point of X § is a limit point of E or a point of E (or both)
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Math 521 - 2/19

  • Feb 19, 2018
  • Shawn
  • Math 521
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Metric Space • Definition ○ A set X of points is called a metric space if ○ there exists a metric or distance function d(p,q):X×X→R such that § Positivity □ d(p,q)0 if p,q∈X and p≠q □ d(p,p)=0 for all p∈X § Symmetry □ d(p,q)=d(q,p) for all p,q∈X § Triangle Inequality □ d(p,q)≤d(p,r)+d(r,q) for all p,q,r∈X • Example 1 ○ X=Rk ○ d(p ⃗,q ⃗ )=|p ⃗−q ⃗ | ○ If k=1, this is just standard numerical absolute value ○ and d is distance on the number line • Example 2 (Taxicab metric) ○ X=R2 ○ d((p_1,p_2 ),(q_1,q_2 ))=|p_1−q_1 |+|p_2−q_2 | where p_1,p_2,q_1,q_2∈R ○ Is this a true metric space? ○ Positivity § Clearly d((p_1,p_2 ),(q_1,q_2 ))≥0 since it is a sum of absolute values § Suppose d((p_1,p_2 ),(q_1,q_2 ))=0 □ |p_1−q_1 |+|p_2−q_2 |=0 □ |p_1−q_1 |=−|p_2−q_2 | □ {█(|p_1−q_1 |=0@|p_2−q_2 |=0)┤⇒{█(p_1=q_1@p_2=q_2 )┤ □ i.e. (p_1,p_2 )=(q_1,q_2 ) § Suppose (p_1,p_2 )=(q_1,q_2 ) □ d((p_1,p_2 ),(q_1,q_2 ))=|p_1−q_1 |+|p_2−q_2 |=|0|+|0|=0 § Thus d((p_1,p_2 ),(q_1,q_2 ))=0 iff (p_1,p_2 )=(q_1,q_2 ) ○ Symmetry § d((p_1,p_2 ),(q_1,q_2 ))=|p_1−q_1 |+|p_2−q_2 | § =|q_1−p_1 |+|q_2−p_2 |=d((q_1,q_2 ),(p_1,p_2 )) ○ Triangular Inequality § d((p_1,p_2 ),(r_1,r_2 ))+d((r_1,r_2 ),(q_1,q_2 )) § =|p_1−r_1 |+|p_2−r_2 |+|r_1−q_1 |+|r_2−q_2 | § =(|p_1−r_1 |+|r_1−q_1 |)+(|p_2−r_2 |+|r_2−q_2 |) § ≥|p_1−r_2+r_1−q_1 |+|p_2−r_2+r_2−q_2 | by Triangle Inequality of R § =|p_1−q_1 |+|p_2−q_2 | § =d((p_1,p_2 ),(q_1,q_2 )) Definition 2.17 • Interval ○ Segment (a,b) is {x∈Raxb} (open interval) ○ Interval [a,b] is {x∈Ra≤x≤b} (closed interval) ○ We can also have half-open intervals: (a,b] and [a,b) • k-cell ○ If a_ib_i for i=1,2,…,k ○ The set of points x ⃗=(x_1,x_2,…,x_k ) in Rk ○ that satisfy a_i≤x_i≤b_i (1≤i≤k) is called a k-cell • Ball ○ If x ⃗∈Rk and r0 ○ the open ball with center x ⃗ with radius r is {y ⃗∈Rk│|x ⃗−y ⃗ |r} ○ the closed ball with center x ⃗ with radius r is {y ⃗∈Rk│|x ⃗−y ⃗ |≤r} • Convex ○ We call a set E⊂Rk convex if ○ λx ⃗+(1−λ) y ⃗∈E, ∀x ⃗,y ⃗∈E, 0λ1 ○ i.e. All points along a straight line from x ⃗ to y ⃗ and between x ⃗ and y ⃗ is in E • Example: Balls are convex ○ Given an open ball with center x ⃗ and radius r ○ If y ⃗,z ⃗∈B, then |y ⃗−x ⃗ |r and |z ⃗−x ⃗ |r ○ |λz ⃗+(1−λ) y ⃗−x ⃗ | ○ =|λz ⃗+(1−λ) y ⃗−(λ+1−λ) x ⃗ | ○ =|λz ⃗−λx ⃗+(1−λ) y ⃗−(1−λ) x ⃗ | ○ ≤|λz ⃗−λx ⃗ |+|(1−λ) y ⃗−(1−λ) x ⃗ | by Triangle Inequality ○ =λ|z ⃗−x ⃗ |+(1−λ)|y ⃗−x ⃗ | ○ λr+(1−λ)r=r ○ Thus |λz ⃗+(1−λ) y ⃗−x ⃗ |r ○ i.e. λz ⃗+(1−λ) y ⃗∈B Definition 2.18 (a) Neighborhood (b) Limit point (c) Isolated point (d) Closed (e) Interior point (f) Open (g) Complement (h) Perfect (i) Bounded (j) Dense
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Math 521 - 2/16

  • Feb 19, 2018
  • Shawn
  • Math 521
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Set-Theoretic Operations • Set theoretic union ○ ⋃24_(n=1)^∞▒A_n =A_1∪A_2∪A_3∪⋯ • Set theoretic intersection ○ ⋂24_(n=1)^∞▒A_n =A_1∩A_2∩A_3∩⋯ • Indexing set ○ ⋃8_(α∈A)▒E_α , where ○ A is an indexing set ○ E_α is a specific set that depends on A • Example ○ Let A={x∈R0x≤1} ○ Let E_α={x∈R0xa} ○ Then⋃8_(α∈A)▒E_α =(0,1) and ⋂8_(α∈A)▒E_α =∅ Theorem 2.12 • Statement ○ Let {E_n }_(n∈N be a sequence of countable sets, then ○ S=⋃24_(n=1)^∞▒E_n is also countable • Proof ○ Just like the proof that Q is countable ○ E_n={〖x_n〗_k }={〖x_n〗_1,〖x_n〗_2,〖x_n〗_3,…} ○ ■(x_11&x_12&x_13&x_14&…&@x_21&x_22&x_23&⋱&&@x_31&x_32&⋱&&&@x_41&⋱&&&&@⋮&&&&&) ○ Go along the diagonal, we have ○ S={x_11,x_21,x_12,x_31,x_22,x_13…} • Corollary ○ Suppose A is at most countable ○ If for α∈A, B_α is at most countable, then ○ T=⋃8_(α∈A)▒B_α is also at most countable Theorem 2.13 • Statement ○ Let A be a countable set ○ Let B_n be the set of all n-tuples (a_1,a_2,…a_n ) where a_k∈A for 1≤k≤n ○ And a_k may not be distinct, then B_n is countable • Proof ○ We proof by induction on n ○ Base case: n=2 § ■((a_1,a_1 )&(a_1,a_2 )&(a_1,a_3 )&(a_1,a_4 )&…&@(a_2,a_1 )&(a_2,a_2 )&(a_2,a_3 )&⋱&&@(a_3,a_1 )&(a_3,a_2 )&⋱&&&@(a_4,a_1 )&⋱&&&&@⋮&&&&&) § Here a_i are all the elements of A with possible repetition ○ Now assume for n=m (m≥2) § The set of m-tuples (a_1,a_2,…a_m ) are countable § Now we treat the (m+1)\-tuples as ordered pairs § (a_1,a_2,…a_(m+1) )=((a_1,a_2,…a_m ),a_(m+1) ) § By n=2 case, the set of (m+1)\-tuples is still countable Theorem 2.14 • Statement ○ Let A be the set of all sequqnecse whose digits are 0 and 1 ○ Then A is uncountable • Proof: Cantor
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