《点集拓扑讲义》第四版(熊金城)课后习题答案

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$ 1 ?fi? (1)

§1.1 $3 (1)

§1.2 $#$ (1)

§1.3§fi (5)

§1.4§'§fi (6)

§1.5$‰ (6)

§1.6*t%#$ (9)

§1.7a

.,fla

.,

9

§1.8???3 (10)

$2?@fl67fl?:$‰ (12)

§2.1<67fl?:$‰§2.2@fl67fl?:$‰ (12)

(15)

§2.3=>fl=>fi (16)

§2.4fl,‰,‰￠ (16)

§2.5?C,4E (18)

§2.6fl? (21)

§2.7@fl67?§fi (23)

$ 3 ? L67,M67 (26)

§3.1?67 (26)

§3.2($)L67 (28)

§3.3M67 (31)

$ 4 ??N? (35)

§4.1?N67 (35)

§4.2?N?$‰RS T$ (38)

§4.3?N y@ (40)

§4.4C?N67§4.5‰Z?N67 (41)

(42)

$5?$§a.??3 (44)

§5.1$—fl$=a.??3 (44)

§5.2ay67 (46)

§5.3L i nd e l o¨

f67 (47)

$ 6 ?y$??3 (49)

§6.1T,T,H a u z d o r f67 (49)

§6.2fl^,flfi,T,T67 (51)

§6.3U r y z o hn fl3aT i e t z e flfl?3 (53)

I

?fl

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§6.4yK fl^67,T y s h o n o f67 (54)

§6.5y$??3fl?67,( L67aM67 (55)

§6.6 (57)

$ 7 ?hi? (58)

§7.1hi67 (58)

§7.2hi?fly$??3 (59)

§7.3n‰fi?67R?

hi ? (60)

§7.4fi?hi??t%7 §fi (61)

§7.5<67? hi? (63)

§7.6Chi67,fihi67 (64)

$ 8 ?yq <67 (66)

§8.1 (66)

§8.2<67yq?fl hi?,B a i r e?3 (67)

$ 9 ?L67 (69)

§9.1

* $?"L (69)

§9.2L67 (69)

§9.3aL

@fl?u (72)

§9.4T y s h o n o f v L?3 (74)

§9.5@fl67flfifl ?z$ (75)

$ 10 ?$‰67 (76)

§10.1Afi~?@fl (76)

§10.2—i~?

§10.2hi fl@fl (76)

? 1 ?

fi÷

§ 1.1

$

1.1 ?

flfifi

???:

‰‰

%6

?

‰‰

%??

?‰‰

7$￠?§fi ?‰

‰

7$??§fi ?

(1) ?={ s |s c Z ,?%flfl y c Z

,? s=2y };

(2) B={2};

(3) C ={s |s c ?,% s =1};

(4) D ={ s |s c ?,% s fi ?.};

(5) E ={s |s c Q ,% s =2}.

?:B =DC ?C Z ,C =E =?,C ,E %

B ??,B ,D %

??.

1.2

??fi ?fi §fifi

fl ?fl ??.

(1) ?={ ?} ; (2) ?c { ?}; (3) ? C { ?}; (4) ? ={?};

(5) ? c {{?}}. ?:(2),(3) fifl ?. 1.3 ? ? ,? ,…,?

?%

,%? n “ 1.??:?$ ? C ? C …C ?

C ? C ?

^

? =? =… =?

?:?fi ? C ? ,i =1,2,…,n –1,? C ? ,$§?,? C ? C ? C ? ,% ? C ?

,? ? ? =? ,i

=1,2,…,n –1. ? ? =? =… =? . 1.4

? X ={a ,b ,c } .fi { X

￥(X ) .

?:￥( X ) ={?,{ a },{b } ,{ c } ,{ a ,b },{a ,c },{b ,c },{a ,b ,c }} . 1.5 ? X fi $ n KA fl ??

y?fifi .X

￥( X ) ?$:?KA fl ??

y??

?:￥( X )}

y?$ 1

,C

KS ,C

K?$QK y? ,…,C

K?$ n –1 K ,C

K?$ n Ky? ( X $?),? ￥( X ) ?$ 1 +C +C +… +C =2

KAfl??

y?.

§ 1.2

$ $

2.1 ?,B ,C ?

% ,??:

(1) ?C ?U B,?t ?3 ?fi B ;

(2) @ ?C B,^$ ?U CC BU Ca ?fi CC Bfi C ; (3)

@ ?C B ,^ B –( B –?) =?.

?:(1) ?3,(2) ?fi ?C B ,?? ?=?fi B ,B =?U B ,flfi B U C =( ?U B ) U C =B U (?U C ) 3 ?U C

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?fi C=(?fi B)f i C=?f i(B fi C)C B fi C

(3) ?fi?C B,^s c B–(B–?)fi%?fi s c B%s?B–?fi%?fi s c?,?B–(B –?)=?.

2.2?X fi ?,?,B§?fi%,??:

;

(1) ?–B=?fi B′

(2) ?–B=(?U B)–B=?–(?fi B);

=?;

=B,B′

(3)@ ?U B=X,?%?fi B=?,^$?′

(4)(?–B)f i(?–B)=(?fi?)–(B U B).

,fi%?fi s c?fi B′

,???:(1)s c?–B,fi%?fi s c?,s?B fi%?fi s c?,s c B′

.

?–B=?fi B′

=?–B

)=?fi B′

)U(B fi B′

=(?fi B′

(2)(?U B)–B=(?U B)f i B′

)=?fi B′

=?–

)U(?fi B′?–(?fi B)=?f i(?fi B)′

)==(?fi?′

=?f i(?′

U B′

B??,?–B =( ?U B)–B =?–( ?fi B)

=B,

=X–?=(?U B)–?=B–?=B,?fl?′

(3) ?fi?fi B=?,??B–?=B,?′

=?.

)′

=(?′

flfi B′

fi B′

)=(?fi

)=(?fi?)f i(B′

(4)(?–B)f i(?–B)=(?fi B′

)f i(?fi B′

=(?fi?)–(B U B).

?)f i(B U B)′

2.3??,?,…,?,B?%,%?n fifl‰..??:

(1) y?

Bfi (U?) =U ( B fi ?)

BU ( fi ?) =fi ( BU ?)

(2) D e M o r g a n

B–(U?)=fi(B–?)

B–(fi?)=U(B–?)

?:(1)s c B f i(U?)e s c B%s cU?e flfl i,?s c B%s c?e flfl i,? s c Bfi ?escU ( Bfi ?)

??Bfi ( U?) =U ( Bfi ?)

s c B U(fi?)e s c B?s cfi?e s c B?s c?(i=1,2,…,n)e?k?i,?s c B?s c?e?k?i,s c B U?e s cfi(B U?)

?? BU ( fi ?) =fi ( BU ?)

(2)s c B–(U?)e s c B%s?U?e s c B%s??(i=1,2,…,n)e?k?i,s

c B%s??e?k?i,s c B–?e s cfi(B–?)

??B–(U?)e=fi(B–?)

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s c B–(f i?)e s c B%s?fi?e s c B%flfl i,s??e flfl i ,s c B%s??e flfl i ,s c B–?e s cU(B–?)

??B–(f i?)e=U(B–?)

2.4??,?,…,?

?%,%?n fifl‰..??:

U?–(f i?)=U(?–?)(?=?)

?:@?X=U?fi .÷?=?,U?–(f i?)=U(?–(f i?))=U(U(?–?))3U(?–?).

fl—fi?,???–?C(?–?)U(?–?)

??k,@ s c?–?,%s c?,s??,^fi s c?

,s c?–?,fi s??

,s c?–?,??$:s c(?–?)U(?–?),flfi:

?–?C(?–?)U(?–?)C(?–?)U(?–?)U(?–?)C…C(?–?)U(?–?)U…U(?–?)CU(?–?)

??, U ( ?–?) CU ( ?–?)

?U?–(f i?)=U(?–?)(?=?)

2.5??aB%Q

K .?$?fl B??}?=B

?=B=(?–B)U(B–?)

??

:??}#$?fl??‰?3,%:?$?,BaC?

%,^

(1) ?=B=B=?;

(2) ?=?=?;

~ ~

(3) flfl—K?,??=?=?;

(4)(?=B)=C=?=(B=C).

?:(1),(2),?3.

~ ~

(3)$?=?,^?=?=(?–?)U(?–?)=?

(4)(?=B)=C=((?–B)U(B–?))=C=(((?–B)U(B–?))–C)U(C–((?–B)U(B–?)))=(((?–B)–C)U((B–?)–C))U(C–((?U B)–(?fi B)))= (?–(B U C))U(B–(?U C))U(C–(?U B))U(?fi B fi C)=(B–(?U C))U(C –(?U B))U(?–(B U C))U(?fi B fi C)=(B=C)=?=?=(B=C).

2.6??n K?,?,…,??‰?,?,}÷?#$,?:?flfi2KA fl??;?%??$n K,y??‰?,?,}÷?#$??flfi2KA fl??.[

?:? fl??N KQ Q fi??‰?,?,}÷?#$?:?flfi2KA fl??.% ,fl?, ?,…,??‰÷?#$flfi?fi{N=2–1KQQ fi?B,B,…,B,?

%??—??a$B,B,…,B?‰÷?#$${.?%?,?,…,??‰÷?#$flfi*fl B,B\–2,…,B?‰÷?#$flfi*??.k‰Q K? ? fl?%?$?$—Cy

??.N

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‰?k??‰?

yfl,‰{?fl$?$=Cy ?????FL?Q?f.]?:yQ???

$—?,??N KQ Q fi? ?‰?,?,}÷?#$?:?flfi2KA fl?? .

?fi,N‰?#$,N K QQ fi? ?:?flfi

+C+…+C=2–1

KA fl??,fi?#$?§fl 6,}#$fifl$§fl ?fl?,‰?y fi f$—

? ??.

$=?,??k n K ?‰?,?,}÷?#$?:?flfi2–1KA fl?? .

fi E=fi ?

E=?fi?f i…f i?f i…f i?i=1,2,…,n

E=?f i ?f i…fi ?fi…fi ?fi…fi ?i

E=?f i ?f i…fi ?fi…fi ?fi…fi ?fi…fi ?i

……

E=?j?i,i,…,i

%??$?$?,E

K Y:fi1,E

K Y:fi C,E

K Y:fi C,…,E

K Y:fi C,?y ?K Y:fi

%y f i

U?

1 +C+C+… +C=2–1

÷ B =E,B=E–B,B=E–B,…,B=E–B,

B =E–(U B),B=E–(U B),…,B=E–(U B),

……

B=?–(U B),B=?–(U B),…,B=?–(U B),

B,B,…,B Y:$2–1KQ Q fi?,% —? a

$B,B,…,B‰ ,?,

}÷?#$$$,%??,?,…,?‰,?,}÷?#$flfi)fl B,B,…,B‰‰÷ ?#$flfi ),?.

$$—?12 B,B,…,B‰ ,?,}÷?#$flfi fl? 4 K Y:fi 2,fl

fi?,?,…,?‰,?,}÷?#$Y:7flfi2KA fl,?4.

9,fi?={(e,…,1,…,e):e=0;1,k?i},i=1,…,n,E=fi?={(1,1,…,1)} E=?fi?f i…f i?f i…f i?={(1,…,e,…,1):e=0;1}i=1,2,…,n

E=?fi?fi…fi?fi…fi?fi…fi?={(1,…,e,…,e,…,1):e=0;1}i

E=?f i ?f i…f i ?fi…f i ?fi…f i ?fi…f i ?={(1,…,e,…,e,…,e,…, 1):e=0;1}i

……

·

4·

C

E=?={(e,…,1,…,e):e=0;1}j?i,i,…,i

fi B=E={(1,1,…,1)},

B =E–B={(0,1,…,1)},B=E–B={(1,0,1,…,1)},

…,B=E–B={(1,1,…,1,0)},

B=E–(U B)={(0,0,1,…,1)},

B =E–(U B)={(0,1,0,1,…,1)},…,

B–(U B)={(1,1,…,1,0,0)},

……

B=?–(U B)={(1,0,…,0)},

B=?–(U B)={(0,1,0,…,0)},…,

B=?–(U B)={(0,…,0,1)}

‰fi=fi2–1KA fl ,??A B,B,…,B,Q ?{$‰2–1

K?A E‰,?,} #$FG flfi2KA fl ,?4.

H fl,J$n K4,y ‰ ,?,}÷?#$F7flfi2KA fl,?4.

§1.3fi

3.1K X={a,b},Y={c,d,e}.L fi‰X xY%$fi O.

P:X xY={a,b}x{c,d,e}={(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}

3.2K XaY

R%4.?U:??X k???,BaY k??C,D,$:

(?U B)x(C U D)=(?x C)U(?x D)U(B x C)U(B x D)

( ?fi B)x(Cfi D)=( ?x C)fi( BxD)

(X x Y)–(?x C)=((X–?)x Y)U(X x(Y–C))

(?–B)xC=(?x C)–(B x C)

3.3K X={a,b,c},Y={d,e,f,g};R={{a,d},{a,e},{b,f}}.÷?={a,c},B={d,e, g}.L[R(?),R(B),R\],aR?$].

P:R(?)={d,e},R(B)={a}

R a n g e R={d,e,f},D o m a i n R={a,b}

3.4K R

fifl4X

fi4Y?

§fi.?U:??k c?,B C X,§fi R(?)f i R(B)=R(?fi B)fi f

}y ijk fl%:??kc s,y c X,@ s?y,R({s})fi R({y})=?.

?:}y?:K x c R(?)f i R(B),%x c R(?),x c R(B),flfl s c?,y c B ,s R x,y R x,@ s?y,R({s})fi R({y})=

?,x c R({s})fi R({y})$$.?s=y,flfi x c R(?fi B), %R(?)f i R(B)C R(?fi B

),?w1.3.2(2)$R(?)f i R(B)3R(?fi B).H fl$R(?)f i R(B)=R( ?fi B)

ij?:K s,y c X,s?y,$R({s})fi R({y})=R({s}f i{y})=R(?)=?

3.5K X,X,X%÷K

4.fl §?$X x X x X=X x X x X?

P:(1)fi X(i=1,2,3)?Y}$—K fi X x X x X=X x X x X=?.

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