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)
