Math 521 - 2/26

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_ic 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 ̅