8/14/2019 Khong Gian Metric Khong Gian Topo KTHP Toan2007AB De1-091223

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trng h ng thp THI KT THC HC PHN s 1 Hc phn: Khng gian mtric-Khng gian tp

Thi gian: 120 phtCho cc lp: Ton 2007ABS n v hc trnh: 3

Bi 1. (3 im) Vi mi x, y X, t d(x, y) =

0 nu x = y,2009 nu x = y. Chng minh

rng:(a) d l mt mtric trn X.(b) (X, d) l mt khng gian mtric y .

Bi 2. (1 im) Chng minh rng trong mt phng R2

vi mtric clit th elip(E):

x2

a2+

y2

b2= 1 l mt tp compc.

Bi 3. (1 im) Chng minh rng khng gian con ng (A, dA) ca khng gianmtric y (X, d) l khng gian mtric y .

Bi 4. (3 im) Hy ch ra (khng cn chng minh) 3 tp khc nhau trn tps t nhin N sao cho (cn chng minh) c 1 tp l Hausdorff v 1 tp lkhng Hausdorff.

Bi 5. (1 im) Chng minh rng, vi tp thng thng, th R ng phi vikhng gian con (0, 1).

Bi 6. (1 im) Chng minh rng, vi tp thng thng, th R = Q.

HTTh sinh KHNG c s dng ti liu khi lm bi

8/14/2019 Khong Gian Metric Khong Gian Topo KTHP Toan2007AB De1-091223

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P N

Bi Ni dung im1 (a) Kim tra 3 tin ca mtric

d(x, y) 0, d(x, y) = 0 x = y 0,5

d(x, y) = d(y, x) 0,5x = z: d(x, y) + d(y, z) 0 = d(y, z) 0,5x = z: Khi x = y hoc z = y, suy ra d(x, y) + d(y, z) 2009 = d(y, z).

0,5

(b) Gi s{xn}n l dy Cauchy trong (X, d). Khi , tn ti n0:d(xn, xm) < 2009 vi mi m, n > n0, suy ra xn = xn0 vi min n0. Do xn xn0.

1,0

2 chng minh (E) compc, ta chng minh (E) ng v b

chn.(E) l tp ng: Gi s dy {(xn, yn)}n (E) v (xn, yn)

(x, y). Vx2n

a2+

y2n

b2= 1 nn

x2

a2+

y2

b2= 1. Vy (x, y) (E).

0,5

(E) l tp b chn: V x2 + y2 x2a2

+y2

b2

(a2 + b2) = a2 + b2 nn

(E) b cha trong hnh cu ng tm O bn knh

a2 + b2.Suy ra (E) b chn.

0,5

3 Ta chng minh dy Cauchy {xn}n trong A l dy hi t.

V {xn}n l dy Cauchy trong A nn dA(xn, xm) 0. Suy rad(xn, xm) 0. Vy {xn}n cng l dy Cauchy trong X. V Xl khng gian mtric y nn xn x trong X. V A ngv {xn}n A nn x A. Vy xn x trong A.

1,0

4 C th chn tp th {,N} l tp khng Hausdorff: NuU, V l ln cn ca 0 th U = V = N. Suy ra U V = .

1,0

C th chn tp ri rc P(N) l tp Hausdorff: Nu x = yth chn U = {x} v V = {y} ta c U l ln cn ca x, V lln cn ca y v U

V =

.

1,0

Tp cn li c th chn l {, {1},N}. 1,05 Xy dng mt php ng phi t (0, 1) ln R, chng hn

f(x) = tg(x 2

) vi mi x (0, 1).

1,0

6 Ta ch cn chng minh R Q. Vi mi x R, t xn =[nx]

nta c xn Q v xn x. Vy x Q.

1,0