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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)](https://shawnzhong.com/wp-content/uploads/2018/03/img_5aa0d67746662.png)
![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](https://shawnzhong.com/wp-content/uploads/2018/03/img_5aa0d2d37f657.png)
