Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2

Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2 C II,,,,,,,,,,, 0.2. 1 (Connectivity) 3 2 (Compactness) 6 3 Separating Axioms) 9 4 (Metric Spaces) 10 0.3,,., (S, O): top. sp., A S. A:, ;,,, A: (cpt);,, : T 0, T 1, T 2, T 3, T 4 -sps.,,,,,,,,,,,,

n-dim. Euclid sp. metric, nbd=neighborhood, ball, open, closed, interior pt=point, exterior, boundary, open kernel, closure, adherent pt, accumulation pt, isolated pt, continuous map.=mapping (ft=function) Topological sp. topology, indiscrete (trivial) top., discrete top., nbd=neighborhood system, sub-basis & basis, fundamental system of nbds, 1st axiom of countability, 2nd axiom of countability, separable, dense (1st-countable sp., 2nd-countable sp.,.) open map., closed map., homeomorphism, induced top., subsp., relative top., product top., Connectivity path-connected, connected component, totally disconnected, intermediate value theorem, Compactness compact, finite intersection property, Tychonoff s theorem, Separation axioms Hausdorff sp., regular, normal, Urysohn s lemma, Metric spaces Cauchy sequence, complete, totally bounded, sequentially compact, distance-preserving map, completion, metrization theorem.

,, (S, O): top. sp.;,, S: a set, O P(S) = 2 S :. (S, O ), (A, O A ), (B, O B ), (S i, O i ) (i = 1, 2,...), (S λ, O λ ) (λ Λ): top. sps. R n,,,, Euclid (=Euclid ).,,,.,,,.,,. f : (S, O) (S, O ) f 1 (O ) O, i.e., V S : open, f 1 (V ) S: open. 1 (Connectivity) 1.1., (.), (A, O A ): (connected) def [ B A: of (A, O A ) B = or B = A], O A = O A A S (S, O)., B A: of (A, Q A ) U, V O; B = U A = V c A., =, A S: U, V O; A U V, A U V =, A U, A V., R [a, b], (a, b], (a, b), [a, b) (a < b), B = (a, b) [c, d) (a < b < c < d)., (a, b), [c, d) O B,. 1.2.. f : S S : conti. S: connected f(s) S : connected. f(s), U, V O ; f(s) U V, f(s) U V =, f(s) U, f(s) V. S f 1 (f(s)) f 1, f S, A B A, A, B. A. B: A:.,, U O, A U = A U =., A U x A U,, U A, A U =. A, B, U, V A B, U, V,.,. 1 x A B, x U, A V B V, A V A..

x S, C(x) = C:,x C,. S def x S, x C(x) = {x}. Q Cantor.,. 1.3. C: x. x, 1.1 ( ) (S, O):, f : S R:, x 0, x 1 S; α = f(x 0 ) < β = f(x 1 ) α < γ < β, x S; f(x) = γ. [ ]. γ (α, β); x S, f(x) γ, f 1 ((, γ)) = f 1 ((, γ]) x 0 S, S., f(s) R, α, β f(s), [α, β] f(s). (, S R:, α, β S,, α < γ < β; γ / S S (, γ), (γ, ) S.) 1.4. = 2 ( )., =, x, y S, f f : [0, 1] S;, f(0) = x, f(1) = y., x, y S f, S x,y := f([0, 1]), x,y S S x,y = S.,., R n,.,,. R 2, (0, 1] {0} {1/n} (0, 1] (n 1) A, {0} (0, 1] B, A B B {(0, 0)} = A, A,, B,. 1.5,.

,, x S, U(x): x, U U(x);., R 2, A = {( x, sin 1 ) } ; x > 0 x., A = A {(0, y); 1 y 1} =: S.. (1/π, 0) S f,. t 0,, f(t) y, ±1, 2 2 f(α), f(β),, α, β t 0, f, f(t 0 ) = (0, 0), 1 ( ).. 1.6. f : [0, 1] S; f(0) = (0, 0), f(1) = (1/π, 0), p : S R; (x, y) x. p(f(0)) = 0, p(f(1)) = 1/π > 0, t 0 = max{0 t 1; p(f(t)) = 0} ( ),, 0 t 0 < 1. 0 < δ < 1 t 0 ; t 0 < t < t 0 + δ, f(t) f(t 0 ) < 1/2. t 0 p(f(t 0 + δ) > 0 n 1, p(f(t 0)) = 0 < {( ) 1 {( ) 1 3 1 π} 2 + 2n < π} 2 + 2n < p(f(t 0 + δ)), {( ) 1 {( ) 1 3 1 α, β (t 0, t 0 + δ); p(f(α)) = π} 2 + 2n, p(f(β)) = π} 2 + 2n. f(α) = (p(f(α)), 1), f(β) = (p(f(β)), 1), 2 < f(α) f(β) f(α) f(t 0 ) + f(t 0 ) f(β) 1 2 + 1 2 = 1,. S = A = A {(0, y); 1 y 1}., 1/2 B, S S B, {(0, y); 1/2 < y < 1/2}, S.,. 1.7 S B, C = {(0, y); 1/2 < y < 1/2}, S. C S, O, 0 < δ < 1/2; U = U δ (O) R 2, S U C., S U C S.

2 (Compactness) 2.1.,.,,., def A S: cpt {U λ }: O.C.(= open covering) of A, {U λi } n i=1 ; O.C. of A. U λ O; U λ A, λ i, i = 1, 2,..., n; n i=1 U λ i A.,, A S (A, O A ). 2.2.,,, [ ]...,..,. 2.3,..,.. Hausdorff = T 2 - sp. (= 2, ),,.,,, ( ).,.,,,,.,.., f 1 = f,,. S: cpt, S : T 2, f : S S :. C S: closed cpt (by S cpt), f(c) cpt (by f: conti.) closed (by S : T 2 ). f: closed map. = open map., f 1 : conti..,,. (.) T 4 (= ).,.

- R n n i=1 [a i, b i ]., 2n,, 1, 2n,.,.,. Euclid R n, =, B δ (O) R n,,, 1,. R n,.,,,,,,.. S: cpt, f: conti., f(s) R cpt,, max, min. 2.4 Tychonoff,...,,,. Zorn, Tychonoff,. [Tychonoff ],. S λ S = λ Λ S λ, S C 2 S, C = {C C}. C 1, X = {X 2 S ; X C},,., X Y X, Y = Y = {X ; X Y} = {A S; A X Y},, Y X., Y. Zorn X 0 X. C X 0,, X 0 = {X; X X 0 }. λ Λ, π λ : S S λ, C λ = {π λ (X); X X 0 }, S λ 1, S λ, C λ., x = (x λ ); x λ C λ 1, x X 0., x X 0 X X 0, x X, x U, U X, U, n i=1 π 1 λ i (V λi ); V λ S λ x λ,., x λ π λ (X) V λ π λ (X) π 1 (V λ ) X (V λ x λ ). [Tychonoff ] 1,,.,. Tychonoff,., A λ (λ Λ), A λ. ω, S λ = A λ {ω}, O λ := {, S λ, {ω}, B c ; B < },, (S λ, O λ ) ( )., S := S λ. π λ : S S λ, F λ = π 1 λ (A λ) S ( )., {F λ } λ Λ., S, A λ = F λ. 2.5 (S λ, O λ ),. F λ = π 1 λ (A λ). (, A λ = {ω} c : closed in S λ.) {F λ } λ Λ. ( λ 1,..., λ n Λ, x λi A λi (i = 1,..., n) 1, x λ = ω (λ λ i, i = 1,..., n), x = (x λ ) F λi.)

(Locally Compactness) 2.6,. 1..,. x,, x, x. (x ). 2.7.,.,,. x K: cpt, U: open; x U K o, C: cpt; x C U., T 3, i.e., x U c, U 1, U 2: open; x U 1, U c U 2, U 1 U 2 =, i.e., x U 1 U2 c U., C = U2 c C K cpt. T 3,,. x / C: closed, U: open; x U U C c.,, K: cpt; x K o K C c, U = K o. 2.8. : S: top. sp., S: cpt, ϕ : S ϕ(s) S: isom.; ϕ(s) = S., ( S, ϕ) S. S \ ϕ(s) 1, 1. 1,, (S, O): top. sp., 1 x / S, S = S {x } O ; O S = O,. (, (S, O).) O = O O = {U; U O or U O };, V = S \ F O ; F S., O, S, O, 1 x. O, N (x ) = O, N (x ) S, S = S., S, {U λ } O :, x, S, U λ \ {x } O,., S U λ,. S 1 S,, S S, S, S O.

3 Separating Axioms). 3.1. T 0 -sp., T 1 -sp., T 2 -sp., T 3 -sp., T 4 -sp., Hausdorff sp.,,. T 0 : 2,,. T 1 : 2,,. T 2 : 2,,. T 3 : 1,. T 4 : 2,. Hausdorff= T 2, = T 1 + T 3 -sp., = T 1 + T 4 -sp. T 3, T 4., Wikipedia. 3.2 T 1 1. x S, y x, T 1, U y y: open; x / U y, {x}., {x} c = y x Uy.., [ T 1 + T 4 T 1 + T 3 ]. 3.3 Hausdorff sp.,. 1. (T 2 T 1.), T 2 + T 4 T 3 ( 1 ).,. T 4,, T 3,. (,.) S S 2 {(x, x); x S}. x S, {x}. ( = ). x y (x, y) / U V c U V =., O O(S 2 ), U, V O; U V O,. x y U, V O; x U, y V, U V =, V c x., x U V c, y / V c., T 2. 3.4 T 3,., x S U, V U: x, i.e., x V o V = V U. U c : closed. 3.5 T 4 T 4; F : closed, G: open; F G, U: open; F U U G.. 3.6 (Urysohn),. S T 4. A, B S, [0, 1] f on S, 0 1, i.e., f(a) = {0}, f(b) = {1}.. Λ: [0, 1] 2 m/2 n, n 1, m = 0, 1,..., 2 n., T 4,. r Λ, O(r) O; A O(0) O(1) = B o, r < r O(r) O(r )., f : S R, f = 1 on B = O(1) c, x O(1), f(x) = inf{r Λ; x O(r)},.

4 (Metric Spaces) 4.1 (S, d).,,, d, d = d/(1 + d), X B(X) = B(X, R), f = sup x X f(x) ( ), d(f, g) = f g, (B(X, R), d). 4.2. C I :. C II :. x S, B r (x); r Q +.,, C II,,,, 1 x n,., : {x n} = S U O, x n U, U n : ; U n U., {x n }, B r (x n ), r Q +, n 1., U: open, x U, δ > 0; B δ (x) U, n; d(x, x n) < δ/2., 0 < r < δ/2 r Q 1, B r(x n) U.,, ; T 2 + T 4 T 1 + T 3. 4.3. 4.4,. 4.5,. X, B(X) d(f, g) = f g,. (X, O), C b (X) = C b (X, R) X, C b (X) B(X). 4.6.,. def ε > 0, ε. def..,, 4.7,. f : (S, d) (S, d ) def x, y S, d(x, y) = d (f(x), f(y)), f(x) = f(y) 0 = d (f(x), f(y)) = d(x, y), x = y,. 4.8,.

( S, d) (S, d) completion, i : S S ; ( S, d), i(s) S.,, (( S, d), i).,,,,. (( S, d), i), ((Ŝ, ˆd), î) (S, d), f : S Ŝ ; î = f i. 4.9,. C II + T 2 + T 4. 4.10 C II + T 3 = T 4. 4.11,,. f : S S def c < 1; x, y S, d(f(x), f(y)) c d(x, y).,. 1,,,.,. x, x 2, d(x, y) = 0, i.e., x = y. d(x, y) = d(f n (x), f n (y)) c n d(x, y) 4.12,.,,.