/ 2 ( ) ( ) ( ) = R ( ) ( ) 1 1 1/ 3 = 3 2 2/ R :. (topology)

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1 (set) x X x X x X 2. (space) Hilbert Teichmüller 2 R

2 / 2 ( ) ( ) ( ) = R ( ) ( ) 1 1 1/ 3 = 3 2 2/ R :. (topology)

3 / S O S (O1)-(O3) (O1) S O O (O2) m N, O 1,..., O m O = O 1 O m O (O3) Λ λ Λ O O λ O λ Λ O λ O S O O S S O O (open set) S (topological space) 4 S S (point) (O1)-(O3) (O3) Λ (index set) n Z R 2 (n, 0) 1 B n := { (x, y) R 2 : (x n) 2 + y 2 < 1 } B 2 B 1 B 0 B 1 B 2 n Z B n Λ = Z λ = n a R B a := { (x, y) R 2 : (x a) 2 + y 2 < 1 } B a B 0 B 1 a R B a Λ = R

4 / S 3.2:. 5

5 / 5. S (O1)-(O3) O (O1) S 1 O O (O2) (O3) (O2) O O (O1) (O3) O O O O

6 / 6 O (O1) S O. = = 3 S = {,, } O (S, O) 1. O = {, {, }, {, }, S} 2. O = {, { }, { }, { }, S} 3. O = {, { }, {, }, { }, S}. 5 S = {,,,, } O := {, { }, { }, {, }, {, }, {,, }, S} (S, O) 3.3 (O1)-(O3) (O1)-(O3) (O2) (O3)

7 / R n x, y R n d(x, y) 0 1 r B(p, r) := {x R n : d(x, p) < r} p r (open ball) A R n R n (open set) p A r > 0 B(p, r) A 6 R n O(R n ) R n R n O S = R n, O = O(R n ) (O1)-(O3) (R n, O(R n )) (O1) S = R n (O2) O 1,..., O m O O = O 1 O m O (O1) O O O p O p O j (1 j m) r j > 0 B(p, r j ) O j r 1,, r m r B(p, r) O j j B(p, r) O O O O R 2 B n := { (x, y) R 2 : x 2 + y 2 < 1/n } (n = 1, 2,...) (O3) Λ λ Λ O λ O = λ Λ O λ p O λ Λ p O λ

B(p, r) A 6 R n O(R n ) R n R n O S = R n, O = O(R n ) (O1)-(O3) 3.3.1 (R n, O(R n )) (O1) S = R n (O2) O 1,.

8 / 8 p O λ B(p, r) O λ r > 0 B(p, r) O λ O O (O1)-(O3) R 4 S x, y S 0 d(x, y) (MS1) d(x, y) = d(y, x) (MS2) d(x, y) = 0 x = y. (MS3) z S d(x, y) d(x, z) + d(z, y). d : S S R, (x, y) d(x, y) S (metric, distance) S d (S, d) (metric space) 1. R n d (MS1)-(MS3) (MS1) (MS2) (MS3) R n R n x = (x 1,..., x n ), y = (y 1,..., y n ) d (x, y) = max x i y i 1 i n d (MS1)-(MS3) (R n, d ) 2. I = [ 1, 1] R C 0 (I) f, g C 0 (I) d (f, g) = (C 0 (I), d ) max f(x) g(x) 1 x 1

9 / 9 3. V, d : (x, y) x y, x y 0 V (S, d) p S r B(p, r) := {x S : d(x, p) < r} p r (open ball) A S (S, d) (open set) p A r > 0 B(p, r) A (S, d) O(S, d) (S, d) O (S, d). (O1)-(O3) R n. (R n, d) O (R n, d ) 3.4

10 / 10 (S, O) A S A (closed set) S A O A (interior) A A A A (interior point) A (closure) A A 7 A (boundary) A A A A := A A 3.3: A C 2 A A A 3.3. A S (1) A A = A A = A. (2) A A = A (3) A A = A (4) S z A z A S A S = {,,,, } O (S, O) S A = {,, } A A A

11 / (S, O), (S, O ) f : S S (continuous) S S O O = f 1 (O ) O 8 f : R R ϵ-δ f : R R ϵ-δ f : R R p R ϵ > 0 δ > 0 x p < δ f(x) f(p) < ϵ p R f f. (S, d) (S, d ) (S, d) p S r B(p, r) (S, d ) q S s B (q, s) f : (S, d) (S, d ) p S ϵ > 0 δ > 0 f(b(p, δ)) B (f(p), ϵ) p S f f R d(x, y) = d (x, y) = x y ϵ-δ

1 f : R R ϵ-δ f : R R p R ϵ > 0 δ > 0 x p < δ f(x) f(p) < ϵ p R f f.

12 / : B (f(p), ϵ) B(p, δ) (S, O) (S, O ) O, O f : (S, O) (S, O ) p S f(p) O O p O O f(o) O p S f f (S, O) (S, O ) (S, d) (S, d ) f : (S, O) (S, O ) f O O f 1 (O ) O f

13 / 13. = p S f(p) O O O O := f 1 (O ) O p O f(o) O f = 9 O O p f 1 (O ) f(p) O O p O f(o p ) O O := p f 1 (O ) O p (O3) O O p O p p f 1 (O ) f 1 (O ) O f(o) = p f 1 (O ) f(o p ) O O f 1 (O ) O = f 1 (O ) O f (S, O), (S, O ) f : S S (homeomorphism, topological map) 1. f 1 : S S 2. f f 1 (S, O) (S, O ) (homeomorphic) f : S S f O O = f 1 (O ) O f 1 O O = f(o) O 9

) O = f 1 (O ) O f 3.5.3 (S, O), (S, O ) f : S S (homeomorphism, topological map) 1.

14 / 14 f(o) = O S S f (O1)-(O3) S f S (S, O) (S, O ) f f (S, O) U S (relative topology) O U := {O U : O O} U (U, O U ) R n 1. (mapping) f : X Y f X Y function f a. X 2 Y 1 b. X 1 Y a X f : X Y = R, kg 5.0kg Y (correspondence)

15 / : 3.6: 4. X Y

1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete sp

1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete space T1 T2 =, X = X, X X = X T3 =, X =, X X = X 2 X