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ρ (x
n , y n ) } " " ! % {x n } " * {y n } " ! lim n →∞ ρ (x n , y n ) = 0 {y n } " ! c & {x n } {y n } " ! lim n →∞ ρ (x n , y n ) = 0 + " {x n } ∼ { y n } ' 3 ! 5 " ( " ' {x n } , {y n } , {z n } , {t n } " ! {x n } ∼ {y n } {z n } ∼ {t n } lim n →∞ ρ (x n , z n ) = lim n →∞ ρ (y n , t n ) . 9 # 2 * ) 2 ' X x 0 r > 0 X x 0 r {x ∈ X : ρ (x, x 0 ) < r} & B (x 0 , r ) ' x 0 ∈ X r > 0 ρ (x, x 0 ) ≤ r ! " ! x ∈ X & B [x 0 , r ] x 0 r ρ (x, x 0 ) = r " ! ! x ∈ X & S [x 0 , r ] x 0 r x 0 ε > 0 B (x 0 , ε ) ! x 0 ε − - O ε (x 0 ) X M & x ∈ X * ε > 0 ! O ε (x) ∩ M = ∅ + x M " M & ! & M [M] M * x ∈ X O ε (x) ; M ! & x ∈ X M & M & x ! ε > 0 + O ε (x) ∩ M = {x} x M & $ & " * X M & ! M = [M] + M ' ! & " ! * x ∈ M ! ε > 0 + O ε (x) ⊂ M x M & * & ! ! _ ( ! & & * `" ( ! ! ρ (x, z) ≤ max {ρ (x, y) , ρ (y, z)} " (X, ρ) .! & & "" # "" # "" M X \M R # 2 * ) 2 --4 + ! & & #`#"#``" " 2 ! & ; "" ( = ! " c ! "# X = R + ρ (x, y) = |x − y| * B (1, 5) = {x ∈ [0, ∞) : |x − 1| < 5} # a B (3, 4) = {x ∈ [0, ∞) : |x − 3| < 4} ` b ! r 2 = 5 > r 1 = 4 [0, 6) = B (1, 5) ⊂ B (3, 4) = [0, 7). R 3 & ρ (x, y) = 3 i =1 sign | x i − y i | 3> # %5 5 # 4 5 % 4 !5 ` ! ! "# Z # " (0, 1, 2) ! " & ! (0, 1, 2) ! & #`# ! 5 Z % " (0, 1, 2) ! " & + ! ! & #`# ! 5 Z ` f R 3 (0, 1, 2) ! & ! & "$"# \ & ! & & \ M ! [M] = M 2 ! M & & x ∈ M " ε ∈ (0, 1) ! O ε (x) = {x} M \ M ! & ! & & R (a, b) ! & x ∈ (a, b) ε = min{x − a, b − x} ! O ε (x) ⊂ (a, b). [ ( − ∞, a) & ! * x ∈ (−∞, a) ! ε = a−x O ε (x) ⊂ (−∞, a). (b, ∞) & ! .! & ! ( − ∞, a) ∪ (b, ∞) & ! ! #``" ! & ! & [a, b] & $ K + K := {z = x + y : x, y ∈ K} = [0, 2] = & K x = ∞ i =1 ε i 3 i y = ∞ i =1 δ i 3 i ' ε i δ i 0 2 * x = ∞ i =1 ε i 3 i = 2 ∞ i =1 a i 3 i y = ∞ i =1 δ i 3 i = 2 ∞ i =1 b i 3 i a i b i 0 1 x + y = 2 ∞ i =1 c i 3 i , c i = a i + b i c i − 0, 1 2 _ ( x + y, x ∈ K, y ∈ K [0, 2] \ K + K = [0, 2] () * +, * *+- + # - % * 8 ` ! + " & R 2 & x = (x 1 , x 2 ) , y = (y 1 , y 2 ) ! 5 ρ 1 (x, y) = |x 1 − y 1 | + |x 2 − y 2 | ; 5 ρ 2 (x, y) = max (| x 1 − y 1 | , | x 2 − y 2 | ) 4 !5 ρ 3 (x, y) = / (x 1 − y 1 ) 2 + (x 2 − y 2 ) 2 ; 5 ρ 4 (x, y) = sign |x 1 − y 1 | + sign |x 2 − y 2 | 0 = (0, 0) # ! B (0, 1), & B [0, 1] S [0, 1] ! ' = ! R ∗ = (−∞, +∞) ∪ {−∞} ∪ {+∞} 5 ρ 1 (x, y) = | x − y | 1 + | x − y | 4 5 ρ 2 (x, y) = ## ## x 1 + |x| − y 1 + |y| ## ## * 0 < r < 1 B (−∞, r) ; B (∞, r) & ( ±∞ r ! ! " . M ⊂ (X, ρ) & diamM = sup x,y ∈M ρ (x, y) " , - 5 diamB (x 0 , r ) ≤ 2r 4 5 diamB (x 0 , r ) < 2r 4 !5 diamB (x 0 , r ) = diamB[x 0 , r ] c / .! " ! & & & & " R 2 (0, 1) (0, −1) OX (−1, 1) & M M & [ - ρ (x, y) = / (x 1 − y 1 ) 2 + (x 2 − y 2 ) 2 ; x = (x 1 , x 2 ) , y = (y 1 , y 2 ) ∈ M. , (0, 0) # ! & & & 2 & ! & ! \ ! \ )! ! ) )! ! ) $ F 1 F 2 & & ( F 1 ∩ F 2 = ∅ , G 1 ⊃ F 1 G 2 ⊃ F 2 ! & + G 1 ∩ G 2 = ∅. % . & ! " & R ! & ! " ' (−∞, a) , (a, ∞) , (−∞, ∞) (a, a) = ∅ & + ' R & & ( Download 1.57 Mb. Do'stlaringiz bilan baham: 
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