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
