Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology) 17
Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N. ( N n, N n. element e, E.) N = {1, 2, 3,...} = {n; n } 0 (integers) Z ( : Ganz Zahlen Z). A B A B ; x A B Z = {0, ±1, ±2, ±3,...} =( N) {0} N x A or x B. 1, (rational numbers) 2 Q. (, R, Q ) Q := Z/N = {r; r = m/n, m Z,n N} = {m/n; m, n Z,n 0} π 2 (real numbers) R x 2 = 1 i = 1,, (comlex numbers) C C = {z; z = x + iy, x, y R} 0.1 2.., [A B], A [B ],. [B A ]. ( ) 2 =, ( ) 2 =. 1, 1/π, 2/ 3, /. 2,.
Basic Math. 2 1 Keywords (set) A, a, [a A a A (element). a A ]. (A a. A a.) a/ A or A a. A = {2, 3, 5} = {a; a =2, 3, 5}, B = {2, 4, 6, 8,...} = {b; b },., (empty set),. (finite set) ; A <. (infinite set) ; A =. (.) A, B, A B or A B A B [x A x B] x A, x B,, A, B. (., B A. B A.. A B x 0 A; x 0 B, i.e., A B A = B A B,A B [x A x B]. A B and A B, A B. X.,, ( ) (power set), 2 X or P(X). 2 2, 3 n X (set). A, B (subset), i.e., A, B X. A B := {x X; x A or x B} : A B (union) A B := {x X; x A and x B} : A B or (intersection) A B =, (disjoint), A B ( ). ( ).
Basic Math. 3 (1) : A B = B A, A B = B A. (2) : A A = A, A A = A. (3) : A (B C) =(A B) C, A (B C) =(A B) C. (4) : A (B C) =(A B) (A C), A (B C) =(A B) (A C). (5) A = A, A =. (6) A B A A B. (7) A B A B = B A B = A. 1.. X {A n } n=1 (a family of subsets),. x n=1 A n n 1; x A n (n x ) x n=1 A n n 1,x A n. A, B A c := {x X; x/ A} A X (complement). A \ B := A B c = {x X; x A, x / B} (difference). 2. De Morgan. ( ) c ( ) c (1) (A B) c = A c B c, (A B) c = A c B c (2) A n = A c n, A n = A c n. 3. f = f(x), g = g(x) R,. n=1 n=1 f g := max{f, g}, f g := min{f, g} 1. (1) {x R; f(x) a} {x R; g(x) a} (2){x R; f(x) a} {x R; g(x) a} 4. (symmetoric difference) A B := (A \ B) (B \ A), (1) A B =(A B) \ (A B) (2) A C (A B) (B C). {A n } n=1 A 1 A 2, (increasing sequence), A n ( A n ), A 1 A 2, (decreasing sequence), A n ( A n ). 5. 1. (1) [ 0, 1 1 ) (2) [ 0, 1 1 ] n n n 1 n 1 (3) n 1 [ 0, 1+ 1 ] n n=1 (4) [ 0, 1+ 1 ) n n 1 n=1 (1) (2) [0, 1), (3) (4) [0, 1],, [HW] 1. (1) n 1 [ 1 n, 2 1 ) n (2) n 1 ( 1 n, 1+ 1 ) n
Basic Math. 4 lim sup n A n = inf sup n lim inf n A n = sup inf A n := n 1 n A n := n 1 A k k n A k k n (upper limit set) (lower limit set), lim A n, {A n } (limit set). n 6.. (1) lim inf A n lim sup A n. (2) (lim inf A n) c = lim sup A c n. n n n (3) A n = lim n A n = A n. n=1 n A n = lim n A n = A n. 7. R {f n (x)},. {x R; f n (x) f(x) 1/k}. k 1 N 1 n N X, Y, X Y := {(x, y); x X, y Y } X Y, (product), (x, y). ((x, y) (y, x), X Y Y X ). {X n } n 1, X n X 1 X 2 := {(x n ) n 1 =(x 1,x 2,...); x n X n,n 1} ( Π n x n n=1 ). P k :(x n ) n x k n X n X k (projection). 1.. (1) X (Y 1 Y 2 )=(X Y 1 ) (X Y 2 ) (2) X (Y 1 Y 2 )=(X Y 1 ) (X Y 2 ). X A = {A t } t I (I (index set), I =[0, 1] ) n=1 x t I A t = A = {A t ; t I} t I; x A t x t I A t = A = {A t ; t I} t I,x A t De Morgan ; ( A t)c = ( A c t, A t)c = A c t. t I t I t I t I {X t } t I, X t := {(x t ) t ; x t X t,t I} ( Π t x t t I ). P s :(x t ) t x s t X t X s.
Basic Math. 5 3 (mapping) X, Y. X x Y y, f : X Y ; x y = f(x), (mapping). f(x) ={f(x); x X} Y f (range). A X, B Y, f(a) ={f(x); x A} f A (image), f 1 (B) ={x X; f(x) B} f B (inverse image) or. f( ) =, f 1 ( ) =. y f(a) x f 1 (B) f : X Y. x A; y = f(x) f(x) B ( x 1. f 1. B,B 1,B 2,... Y, (1) B 1 B 2 = f 1 (B 1 ) f 1 (B 2 ) (2) f 1 (B 1 B 2 )=f 1 (B 1 ) f 1 (B 2 ), f 1 (B 1 B 2 )=f 1 (B 1 ) f 1 (B 2 ) (3) f 1 ( n B n)= n f 1 (B n ), f 1 ( n B n)= n f 1 (B n ) (4) f 1 (B c )=f 1 (B) c 2. f, A, A 1,A 2,... X, (1) A 1 A 2 = f(a 1 ) f(a 2 ) (2) f(a 1 A 2 )=f(a 1 ) f(a 2 ), f(a 1 A 2 ) f(a 1 ) f(a 2 ) (3) f( n A n)= n f(a n), f( n A n) n f(a n) (4) (2) = 3. A X, B Y, (1) f(f 1 (B)) = B f(x), f 1 (f(a)) A (2) f(a f 1 (B)) = f(a) B, f 1 (f(a) B) A f 1 (B) f : X Y, f (injection) or (one to one) y f(x), 1 x X; y = f(x) ( f(x) f 1 : f(x) X; y f 1 (y).) f (surjection) or (onto) f(x) =Y, f(x) Y, i.e., y Y, x X; y = f(x) f (bijection) f, i.e., y Y, 1 x X; y = f(x) f : X X; f(x) =x (identity mapping), I X,id X I,id. f : X Y, g : Y Z, g f : X Z g f(x) =g(f(x)). ; (h g) f = h (g f): h g f. 4. f [f(x) =f(y) x = y] [x y f(x) f(y)]. (.,.) 5. f : X Y, g : Y Z g f : X Z, (1) f, g g f (2) f, g g f (3) f, g g f (g f) 1 = f 1 g 1 (4) g f f, f g (5) g f g, g f
Basic Math. 6,,,,. f : X Y. g : Y X, g f = id X f g = id Y, f, g = f 1. g f = id X f. f(x 1 )=f(x 2 ), g,, f. x 1 = g f(x 1 )=g(f(x 1 )) = g(f(x 2 )) = g f(x 2 )=x 2 f g = id Y f. y Y, y = f g(y) =f(g(y)), x = g(y) x X y = f(x), f. 6. f. ( f 1. [ {.) (1) f : N Z; f(n) =( 1) n n ] k (n =2k +1,k 0) =,. 2 k (n =2k, k 1), [x], x. [f 1 (m) = 2m +1(m 0), =2m (m 1)] (2) N 2 := N N, f : N 2 N; f((p, q)) = p +(p + q 1)(p + q 2)/2. (, n := f((p, q)),k:= p + q 1, n k q.) (3) A N. f : A N; f(a) = {n A; n a} (A, ) (4) Q + := {x Q; x > 0} A := {(p, q) N 2 ; p, q } f : A Q + ; f((p, q)) = p/q. (5) Q + = {r n } n=1. f : N Q; f(n) =( 1)n r [n/2]., X Y,,,,.,., X Y f X Y., X Y ; X = Y, X Y ; X Y.., Z, N n (n 1), Q, N ( ),,. N Z N n Q, i.e., Z = N n = Q = ℵ 0 (:= N ). 4 ε-δ (ε-δ Logic) [ ] s.t.= such that ( ; ( ).)
Basic Math. 7 x I; x a <δ x a <δ x I. ε. L, M. δ ε>0 ( ) ( ). (.) N ε ( ). (.) ε 0 N 0. (.) {a n } α R. ε, N, N n a n α ε., ε>0, N N, n N, a n α <ε. [ lim n a n = α] or[a n α (n )] ε>0, N N; n N, a n α <ε. [ ] N ε N = N(ε) orn = N ε.,,,,!! {a n } α R. [ lim n a n α] or[a n α (n )] ε 0 > 0; N N, n N; a n α ε 0 ε 0 > 0; k N, n k; a n α ε 0 ε 0 > 0; {n k } k 1 ; n k (k ), a nk α ε 0. [ ] N ( k), ε 0. n ε 0. 2 3, k n = n k. (, n,. {n k }.) {n k } ε 0., ε 0 {n k }. [ 1],. 1. S R, (a) (supremum) α = sup S (1) S ( c R; x S, x c),
Basic Math. 8 (i) x S, x α, (ii) ε>0, x ε S; α ε<x ε ( α) (,,,.) (2) S, α =, i.e., sup S =. (b) (infimum) β = inf S (1) S ( d R; x S, d x), (i) x S, β x, (ii) ε>0, x ε S;(β ) x ε <β+ ε (,,,.) (2) S, β =, i.e., inf S =. 2. I =[a, b] f = f(x) (a) f(x) x 0 I (continuous at x 0 ) ε>0, δ>0; x I; x x 0 <δ, f(x) f(x 0 ) <ε. (b) f(x) I (continuous on I) f(x) x 0 I x 0 I, ε>0, δ>0; x I; x x 0 <δ, f(x) f(x 0 ) <ε. (δ = δ(x 0,ε) > 0 x 0 I,ε > 0. ( f.) ) (c) f(x) I (uniform continuous on I) ε>0, δ>0; x, y I; x y <δ, f(x) f(y) <ε. (δ = δ(ε) > 0 ε>0. (x, y.)) 3. S R f n (x) f(x) (a) f n f S (pointwise convergence) f n f on S (or f n f p.w. on S) x S, ε>0, N N; n N, f n (x) f(x) <ε. (N = N(x, ε) > 0 x S ε>0.) (b) f n f S (uniform convergence) f n f on S (or f n f unif. on S) ε>0, N N; n N, x S, f n (x) f(x) <ε. : ε>0, N N; n N,sup x S f n (x) f(x) ε. lim sup f n (x) f(x) =0 n x S (N = N(ε) > 0 ε>0. x.), f S ε 0 > 0; δ>0, x δ,y δ S; x δ y δ <δ, f(x δ ) f(y δ ) ε 0. x δ,y δ, x δ y δ <δ δ x, y,. ( ε 0,, f.) δ>0, n 1,, δ =1/n, x δ,y δ x n := x 1/n,y n := y 1/n,. (.)
Basic Math. 9 5 (Potency) N, Z, Q, R,,,. (Natural numbers, Integers, Rational numbers, Real numbers) X, Y, X Y (equipotent) f : X Y ;, X Y (potency) ( (cardinals)). X X, X, Card X,. =0, {1, 2,...,n} = n. X X (finite set) X < ( X =0 X = ) (infinite set) X X N X = ℵ 0 (aleph zero), (countable)., X ℵ 0 (X ),. (uncountable). Z, N n (n 1), Q.., R. (1) Z, i.e., Z = ℵ 0. [ f : N Z; f(n) =( 1) n n ] = 2 { k (n =2k +1,k 0) k (n =2k, k 1),., [x], x. ( ) (, f 1 (m) ). [f 1 (m) = 2m +1(m 0), =2m (m 1)] (2) Q, R. (a) N 2 := N N. N n (n 1). f : N 2 N; f((p, q)) = p +(p + q 1)(p + q 2)/2,. (, n := f((p, q)),k := p + q 1, n k q.) (b) X A X N, N A, A N. A,., f : A N; f(a) = {n A; n a},. (c) Q + := {x Q; x>0}
Basic Math. 10 A := {(p, q) N 2 ; p, q },, A N. f : A Q + ; f((p, q)) = p/q,. (d) Q N Q +, Q + = {r n } n=1. f : N Q; f(n) =( 1) n r [n/2],. Z N, B, f : B Q; f((p, q)) = p/q. (e) R ( ), (0, 1] R, (0, 1] = {a n } n 1. 10, a n =0.a n1 a n2 a n3 (a ni =0, 1,...,9)., 1=0.99 2 (,,.), { 1 (ann =0, 2, 4, 6, 8) b =0.b 1 b 2 b 3, b n = 2 (a nn =1, 3, 5, 7, 9)., (a ni ) a nn b n b. b (0, 1], b n a nn, n 1,b a n,., R., R. 2. (1) (2) (a) (d), f. f 1., (potency of continuum), R = ℵ. 3.. <a<b<. (1) [0, 1] [a, b], [0, 1) [a, b), (0, 1] (a, b], (0, 1) (a, b) (2) [0, 1] [0, 1), [0, 1) (0, 1), (0, 1) (0, 1], (0, 1] [0, 1] (f :[0, 1] [0, 1) 1/n 1/(n +1),.) (3) R ( 1, 1) (y = 2 π arctan x or x = tan(πy)/2) (4) (0, 1) 2 := (0, 1) (0, 1) (0, 1) ( f :(0, 1) 2 (0, 1), 10,, ),,. R n (n N). X, ( (power set) ) {A; A X} 2 X (. P(X).)., X Y Y X := {f : X Y ; }, Y 2 ( Y = {0, 1}) Y X (= {0, 1} X )=2 X. X x, (f(x) = 1), (f(x) =0)
Basic Math. 11. f(x) =1 x X. f 2 X X A = {x X; f(x) =1},. X Y X < Y f : X Y ; X Y and X Y. Cantor : X < 2 X (= 2 X ), ℵ 0 < 2 ℵ0 = ℵ f : X 2 X ; f(x) ={x},,, X 2 X. X 2 X. 2 X X.. g :2 X X;. B := {a = g(a) X; A 2 X,a= g(a) / A} ( X), b := g(b), b X, b B,, A X; g(b) =b = g(a) / A, g(b) =g(a) g, B = A, b/ A = B,. b/ B, A = B, b B., 2 X X., X < 2 X. (Schröder-) Bernstein : X Y Y X = X = Y., f : X Y g : Y X. y Y, f x X. x X, g y Y., (a) A X a X, Y, X,Y, X,... a ; a b 1 a 1 b 2 a 2 (b) A X X a X,, Y, X,Y, X,..., Y, X, X, Y a, a or a b 1 a 1 b n a n (c) A Y X a X, Y,X,Y,X,...,X,Y, Y, X a, a b 1 a 1 b n (d) X, Y, B,B Y,B X Y., X = A A X A Y ( ), Y = B B Y B X ( ) f(a )=B, f(a X )=B X, g(b Y )=A Y. f,g. ( f(a Y )=B Y, g(b X )=A X, f(a Y ) B Y, g(b X ) A X. f(a Y ) X Y. g(b X ).) h : X Y h = f on A A X, h = g 1 on A Y.
Basic Math. 12 (0, 1) 2 (0, 1). (0, 1) (0, 1) 2.. (a, b) (0, 1) 2, 10, a =0.a 1 a 2 a 3, b =0.b 1 b 2 b 3.,, 9 (, 0.23 = 0.22999 ). f :(0, 1) 2 (0, 1) f(a, b) :=0.a 1 b 1 a 2 b 2,, (0, 1) 2 (0, 1).. 0.101010, [0, 1] [0, 1] 2,, 0.11010101 (01 ), a =0.1 =0.100000, b =0.11111,, a. (,, 0.11191919.)., 0, e.g., x =0.0010320450001 =0.x 1x 2x 3x 4x 5x 6 (x 1 = 001, x 2 = 03, x 3 =2,x 4 = 04, x 5 =5,x 6 = 0001,...) a =0.x 1x 3x 5, b =0.x 2x 4x 6 x (a, b). : ℵ 0 < ℵ, (,,, 1963, Cohen ) A, B, F (A, B) A B B A. X, X (x 1,x 2,...) (x i X).. 4. A, B, C F (A B,C) F (A, F (B,C)). 5. 2 N {0, 1} N = F (N, {0, 1}) R. 6. R Z [0, 1) N R. (.) 7. F (R, R) 2 R. [F (R, R) F (R,F(N, {0, 1})) F (R N, {0, 1}) F (R, {0, 1})={0, 1} R 2 R ] 8. R R N = F (N, R) R, N R. a 0 + a 1 x + a 2 x 2 + + a n x n =0 (n N,a i Z,a n 0), A., A c := C \ A. 9. A. [N = n + a 0 + a 1 + + a n,.] 10. α A Nα A, β A c Nβ A c. 11. A c., R, Q c., [ x 0, Nx 0 := {x = nx 0 : n N},,. R = Q Q c N Q c Nx 0 Q c = Q c. N Q c Nx 0 Q c, ]
Basic Math. 13 6 (Equivalence Relation and Order) X, 2 x, y X (relation) x y, x y. ( X, Y R X Y X Y. x y (x, y) R ). 3, (equivalence relation) : [ ] (reflexive law) [ ] (symmetric law) [ ] (transitive law) x x x y = y x x y, y z = x z, X. [x] :={y X; x y} (equivalence class). 6.1 X = Z, x y x y 2Z,, [x] =2Z if x 2Z, [x] =2Z +1ifx 2Z +1 2. 1. [x],. (1) x y = [x] =[y] (2) x y [x] [y] = x [x], {[x]} x X X. {[x]} x X =: X/, X (quotient set). X/ = {2Z, 2Z +1}. Z/(2Z). (n N Z/(nZ) ) 2. f : X X/ f(x) =[x]. 3. X = R, x y x y Z,, f :[0, 1) R/ ; f(x) =[x]. X, 3, (order) : [ ] (reflexive law) x x [ ] (anti-symmetric law) [ ] (transitive law) x y, y x = y = x x y, y z = x z x y x y, x y. (X, ) (ordered set). x, y X x y y x. [ ]. 6.2,. (1) S X =2 S. A, B X, i.e., A, B S, A B A B. (2) X = R 2 x =(x 1,x 2 ),y =(y 1,y 2 ) X, x y x 1 y 1,x 2 y 2. (X, ) x X ( ) y X; x y (y x),x = y.,. A X; A, [x X A x A, a A, a x], [x X A a A, a x], [ = ( )].. 2,. 4.. [N],.
Basic Math. 14 7 Zorn (Axiom of Choice, Zorn s Lemma) 7.1 3 (1) 1 ( ). (2) Zorn. (3) X. 2. Zermelo {X t } t T, 1, i.e., X X t ; t T,X X t = {x t }. (, {X t } t T, Y t := {t} X t. {Y t }.) X, f :2 X \ { } X; f(a) A.. 1. 2. [ 7.1 ] [ ] X λ (λ Λ). X := λ Λ X λ,,, (X, )., X λ X,. [Zorn ] X, W := {(A, α); A X, α }. 1 W. (A, α), (B,β) W,,. (A, α) (B,β) A B,α = β A. (W, ) ( ). Zorn, (T, ) W: i.e., (T, ) (T, ) W (T, ) = (T, ). T = X, (X, ). [ ] W 0 W. W := S W 0 = S {A;(A, α) W 0} X, ω, x, y W, (A, α); x, y A, xωy xαy, W 0 well-ined, (W, ω) W 0 1. [ ] T ( X, a X \ T 1, T := T {a} = on T, x T, x a (T, ) (T, ) W, T ( T T. T = X. [ Zorn ] (X, )., W ( x X; w W, w x). [, 1,, ( f- ),.] ( ), W X a W, W a := {w W ; w a} (W a ), (W a ) :={x X; b W a,b x} a (a X )
Basic Math. 15., a = min W, i.e.,w a =, (W a ) :=X. min( (W a ) W )=a.,. f :2 X \{ } X; f(a) A. W X : f- W : ; a W, a = f( (W a )). min W = f(x) (a = min W (W a ) =X )., f(x) X a = f( (W a )) a,, f-. W f-, W := W = w W W, W f- ( W X ). (X, ), W, w. w W., (X, ). [ ] W f-,.. (1) W 1,W 2: f- W 1 = W 2 or. [ ] W 1 W 2,. b W 2, ϕ : W 1 W 2 b ;, x W 1,ϕ(x) =x, W 1 = W 2 b. W 1 := {x W 1; ϕ(x) x}, y := min W 1. min W i = f(x), x W 1; x y ϕ(x) =x W 1 y = W 2 ϕ(y). (W 1 y ) = (W 2 ϕ(y) ). y = f( (W 1 y )) = f( (W 2 ϕ(y) )) = ϕ(y), y W 1. W 1 =. W 1 = W 2 b. W 1 = W 2 or W 1 a = W 2 (a W 1). (2) W. [ ] M ( ) W, a M. W a f-, m := min(m W ) ( W f- ; ) m = min M., x M; x m x W x W ; f-., m = min(m W ), x m x/ W, i.e., x W \ W. (1) b W ; W = W b., b = min(w \ W ), x W \ W b x. m W, m b. m b x, x m. m = min M., W. (3) f- W, W = W or W = W a (a W ). [ ] (1) f- W, min W = f(x)., W W, a := min(w \ W ). a f- W. a/ W (1), b W ; W = W b. a W a W. W W W W a W a. b = min(w \ W ), a W \ W b a, W b W a., W a W = W b W a W a, W = W a. f- W, W W, a W ; W = W a. (4) W f-. [ ] a W, a f- W. (3) W a = W a,, (W a ) = (W a ), f( (W a )) = f( (W a )) = a, W f-. [ ] w., i.e., w X; w w W f- W,., := {x X; a W,a x}, w. z := f( ) (f ), W := W {z}, W = W z, = (W z ) f( (W z )) = z, W f-, W f-. w. w W w,. w W.
Basic Math. 16 (1). (X, ), (Y, ), X Y ϕ : X Y ;, i.e., ϕ, ; x 1 x 2 ϕ(x 1 ) ϕ(x 2 ). 7.2 ( ) 2 (X, ), (Y, ) 1. (1) X Y, (2) a X; X a Y, (3) b Y ; X Y b. 2. 7.1 (X, ), (Y, ). X 1 = {a X; b Y ; X a Y b }, X 1 = X or a 0 X; X 1 = X a 0. a X b Y, X a = = Y b, a X 1, i.e., X 1. a X 1 b Y, ϕ : X a Y b. x X a, y := ϕ(x) X x Y y. a X 1 X a X 1. X 1 X, a 0 := min(x \ X 1 ), X a 0 X 1 a 0 / X 1. X a 0 = X 1., X 1 \ X a 0, i.e., a X 1 ; a 0 a a 0 X a X 1 a 0 / X 1. X 1 = X a 0., X 1 X X 1 = X a 0. 7.2 (X, ), ϕ : X X,, x X, x ϕ(x). A := {x X; ϕ(x) x}, A =. A a := min A. ϕ(a) a, ϕ, ϕ(ϕ(a)) ϕ(a), ϕ(a) A ϕ(a) a a. A =. [ 7.2 ] X 1 = {a X; b Y ; X a Y b }, Y 1 = {b Y ; a X; X a Y b }. a X 1 b Y, b Y 1. ϕ : X 1 Y 1 b = ϕ(a) ϕ : X 1 Y 1. 7.1, X 1 = X or a X; X 1 = X a, Y 1 = Y or b Y ; Y 1 = Y b. X 1 = X a Y 1 = Y b X a = X 1 Y 1 = Y b, X 1 a X 1 = X a, a/ X a. (1), (2), (3).. (2), (3), i.e., X a Y, X Y b. ϕ : X Y b Y X a ( X), a/ X a ϕ(a) a, 7.2. (2), (3).. 7.3 ( ) (A, ), a A P (a). (1) P (min A) (2) a A; a min A, b A a, P (b) P (a), a A, P (a). A 0 = {a A; P (a) }. A 0 a 0 := min A 0. (1) a 0 min A, a 0 a A a 0, P (a). (2) P (a 0 ), a 0 A 0. A 0 =.
Basic Math. 17 8 (Set and Topology) 2 (topology),. (neighborhood) (open set),,., (metric),,. [ ] X, O X (O1) O, X O. (O2) G 1,G 2 O= G 1 G 2 O. (O3) G α O(α A) = α A G α O.. A.,. (X, O) (topology space). [ ] X, d : X X [0, + ]; (x, y) d(x, y). (D1) d(x, y) 0; d(x, y) =0 x = y (D2) d(x, y) =d(y, x) (D3) d(x, z) d(x, y)+d(y, z) (X, d) (metric space). R 2, x =(x 1,x 2 ) x = x 2 1 + x2 2, d(x, y) = x y. δ>0, U δ (x) {y R 2 ; d(x, y) <δ} x δ-. δ, U(x) x. x =(x 1,x 2 ) S R 2 U(x); x, U(x) S, U(x); x, U(x) S =, U(x); x, U(x) S, U(x) S c. S,.,. S, (closed set). S, (boundary)., S, (S c )= S. 1.. 2.. (1) {(x 1,x 2 ); x 2 1 + x 2 2 < 1} (2) {(x 1,x 2 ); x 2 1 + x 2 2 =1} (3) {(x 1,x 2 ); x 1 x 2 > 0} (4) {(x 1,x 2 ); x 1,x 2 Q} 3. S, S. x S x S, S. 4. 2. 5. S S S. 6. S S S.