华中师范大学数学专业导读简介.pdf

16116?6?/QKe?6(c,9,O/0 negation,/0 and,/0 or,/XJ,K0 implies /?=?0 if and only if.L P LL%P.L P LL?duP.(L P)(P)(L)LeL%PPKL(L G)(P G)LGL?dGP?d3?oK$ok?1/0 /m0$2?1/0?1/0$X1+2 32=1+(2 32)?1+2 32=(1+2)32.36?$/$0?gSUegS?1,.XA B C DAn)(A)B)C)D.?A B C289$2An)(A B)C?A (B C).”A B”kX”A%B”,”BA?7”,”AB?”;?”A B”n)”AB?d”,AB?7”,A?=?B”?A B C(A B)(B C)?/?”?”L/y.0”:=”L/0X:=3.14159.eA?6X(a)(A B)A B(b)(A B)A B(c)(A B)B A(d)(A B)(A B)(e)(A B)A B289$8?w?N?/0?/08?/?0 eA8xA?KPx A,”xuA”,?xA?KPxA x/A,”xuA”.eA8?fN:?Ng,1,2,3,.?8289$3Q:?Nkn?8R1:?N?88Au?x,UOxA?=Ox AxA=5ATkk=?i1A,B,C,X,Y,Z?L8?=i1a,b,c,x,y,z?L8?L8k=T8?k?5XA=1,2,3,4,5,N=1,2,3,.,n,.,k,5?NPX=x|xk5P.XB=x|x2=2.qeEk?8KEx:P(x)BLEk5P(x)?x?8X?af(x)Ex:f(x)aLE?f(x)ua?x?88AB?PA=B,?=S8Xx:x2=1=1,1.?A,B8XJuA?uB,KAuBAB?f8PA B,AuBBA?PB A.?88P,?8?f8eX8P(X)2X5LX?Nf8?8X?8289$4u?8A,A AoAA?f8eB A,B 6=A,KBA?f8PB&A.?,e(n2.1 A=B A B B A.n2.2 eA BB C,KA C.8?$1?A,B8#?8A B:=x|x Ax B,AB?8(The union of A and B)eAx8K8A:=x|,?x A.(The union of the family A)?=N:=1,2,.P+n=1An=x|n,?x An.2?A,B8A B:=x|x Ax BAB?8(The intersection of A and B).=dAB?8eA B 6=,KAB?eAx8K?8A:=x|,x A.(The intersection of the family A)?=N:=1,2,.P+n=1An=x|n,x An.289$53?A,B88A B:=x A|xB(PAB)AB?8eB A,KA BBuA?8(The Complementof B in A)83?,K?S?9?8o,8X?f8X?88Bc:=X BB?888A,B,AB:=(A B)(B A)AB?(The Symmetric difference of A and B)e8?$5n2.3(1)?A B=B A,A B=B A.(2)(A (B C)=(A B)C,A (B C)=(A B)C.(3)?A(BC)=(AB)(AC),A(BC)=(AB)(AC).(4)/kA (B)=(A B),A (B)=(A B).(5)eAn,Bn?8K+n=1(An Bn)=(+n=1An)(+n=1Bn).(6)A B A A B.(7)eA B(),K A B,AO/eA C(),K A C.289$6(8)eA B(),K A B,AO/eC B(),KC B.y n2.4?A,B?8X?f8K(1)Xc=,c=X.(2)A Ac=X,A Ac=.(3)(Ac)c=A.(4)eA B,KAc Bc.n2.5(de Morgan 1)?X,A8K(1)X A=(XA).(2)X A=(XA).AO/eX?8K(3)(A)c=Ac,(A)c=Ac.y=y(1),3Sx X A,kx X,x A.?,xA,l?x XA,?kx (XA),=X A(XA).L5x(XA),x XA,?x XxA,l?x A,?x X A,L(XA)X A,?(1)2.6 eXY 8kkS(x,y)x X,y Y?8X Y XY?(k?(Cartesian product).eX1,X2,.,Xn,k8K?NkS|(x1,x2,.,xn),1de Morgan(1806-1871)-=3N?7xi Xi,1 i n.nYi=1Xi=X1 X2.Xn:=(x1,x2,.,xn)|xi Xi,1 6 i 6 n.ekx8X,KX?QX?Nv5f()X()?N?f:X?8KQX8?k8=1,2,3,.,nQXnQi=1Xi.?8QX8w?euXz8XU?x,Kf:Xf()=x,?QXq?g,?dke?/Jn0 Jn?C=Xx8K3N?f:X?,f()X.f 8xC?JJnb?zx83J=QX6=.eX=QX,1K(IN?)Q:X XQ(f)=f().Px QX,df()Px,x?1 IAO/=N=1,2,3,.,n,.Yi=1Xi=(x1,x2,.,xn,.)|xk Xk,k N.3N?d?/?0?X?N?K/?0?X3N?8Xg:R2 0,+)X=Br(a)|r 0,a R2(a,r)Br(a).R2?Br(a)=x R2|x a|0,a R2,X=Br(a)|r 0,a R2LR2?Nz?8Af:X R1,f(Br(a)=r28?8?N?3.1?X,Y 8ezx X,ky Y AKAN?(C)eT 5LAKPT:=X Y,x T(x),TlX?Y?N?y=T(x)x3Te?N”(image).xy?”(inverse image).XT?Y T?A X,T(A)=y Y|y=T(x),x AA3N?Te?”8(range).5T()=,?B Y,T1(B)=x X|T(x)B B3N?Te?”85T1()=.eT(X)=Y,=y Y,x X,?y=T(x),KTlX?Y?(surjective onto).eT(x1),T(x2)Y,T(x1)=T(x2),7?x1=x2,KTlX?Y?(injective)eT:X Y Q?q?KTlX?Y?V?1-1N?(bijective).XY m31-1Ad3A?lY?X?N?T1:Y X,y Y,x X,?y=T(x),l?PT1(y)=x,T1T?_N?w,dT1Y?X?1-1N?T:X Y N?Y=R1CTN?Vg?2/N?0-?3N?93.2?a,b,c,d R1,a b,c 0,?N=N()0,?n Nk|an a|0,|1n 0|=1n1=?N()=1+1(1L1?)K?n N()k|1n 0|0,-0=|ba|3,Kd limnan=a,limnan=b,N1(0)0,?n N1(0)|an a|0,?n N2(0)|an b|0.-N=maxN1,N2,?n N,|an a|0,|an b|0.|?n?|x+y|x|+|y|,?n N0 0,?N()0,?n N()kd(an,a)0,PBa(x0)=x|x x0|0,=()0,s.t.?0|x x0|f(x)A|0,=()0,?x B(x0)x0 f(x)B(A).?ye limxx0f(x)3K7?limxx0f(x)=A,limxx0f(x)=B,eA 6=B,K0=|BA|3 0,1=1()0,?x B1(x0)x0 f(x)B0(A),2=2()0,?x B2(x0)x0f(x)B0(B),e-=min1,2,K?x B(x0)x0 f(x)B0(A)B0(B),0|B A|=|B f(x)+f(x)A|B f(x)|+|f(x)A|0,N=N()0,?n N,d(Pn,P0).K4.6 3m:?4?y?Pnm(X,d)?:?3p,q X,?limnPn=p,limnPn=q.ep 6=q,Kdd?51,0 0,l limnPn=p,limnPn=q,N1(0)0,?n N1(0)d(Pn,p)0,?n N2(0)d(Pn,q)0,?n N=maxN1,N20 d(p,q)d(p,Pn)+d(Pn,q)=d(Pn,p)+d(Pn,q)20=2d(p,q)3?g?p=q.4.7?(X,d)m(ek?:8n)X?f8)(a):P?/?0(a neighborhood of P):8Nr(p):=q|d(p,q)0,Nr(p)?(b):P8E?4:(limit point)eP?kuP?E?:(c)eP EPE?4:KpE?:(isolated point)(d)E48eE?4:?:8E0 E.(e):P8E?S:(interior point)e3P?+N?N E,E?NS:?8PE(intE).(f)eE?z:E?S:=E E,KEm8(g)E3X?8(complement)(PEc)Ec=p X|pE.(h)eE48E?z:E?4:=E E0,KE?8(perfect set)(i)e3M:q X,?uk?p Ekd(p,q)M,KE4m16k.8(j)eX?z:E?4:E?:KE3X(EuX)R1?/+0mmR2?/?0m?X?8Qm848XQm848n4.8 m(X,d)?m8y?E=Nr(p),q E,d(p,q)0,?d(p,q)=r h,Nh(q)?s Nh(q),dn?kd(p,s)d(p,q)+d(q,s)0.w,Nr(p)E=p,?Nr(p)vkuP?E?:PE?4:g?n(4.10 k8vk4:4.11(a)R2?8N1(0)=z=(x,y)R2|px2+y2=|z|0N:=Nr(x)Gi,(1 i n).l?N ni=1Gi=H,?Hm8(d)dde Morgan4.139(c)?4.15 n4.14?/k0?U?X3R1Gn=(1n,1n),KGnm8(n=1,2,.).G=+n=1Gn=0m8?484.16 eXmE X,eE0LE?4:?8E=E E0E?4n4.17 eXmE X,K(a)E 48(b)E=E E48(c)E F?E?48Fd(a),(c),EE?48y(a)ep X pE,KpE,pE0,?pk?NE?l?N Ec,?Ecm8l?E48(b)eE=E,K(a)LE48eE48KE0 E,l?E=E E0 E E,?E=E.(c)eF48E F,KE0 F0 F,?E=E E0 F.5K?(?8?L195K?(?8?L38k?L”x A”,=x8A?A=B=AB?8?,A=BL(x A)(x B).?E,?L(c,(?),(3),9)()5?L(statement)8(e?:6$(a)L33)()?/(J0c/0”5?#?/(J0(b)LO,?(J3)()(c)/zx,5P(x)0 P(x,P(x),/3x,5P(x)0 P(x,P(x).Xx(P(x)(y,(P(y)(y=x)L?g/3Xx?5P(x)?ey?v5P(y)?7ky=x0(/3?x,P(x)0)Q?K?)lX?z!xP(x):=x(P(x)y(P(y)y=x).qe?zL(x X)P:=x(x X P(x).(x X)P:=x(x X P(x).(x a)P:=x(x R x a P(x).5K?(?8?L20(x a)P:=x(x R x a P(x).(RL8)|?(limxaf(x)=A):=0,0,x R(0|x a|f(x)A|a)P)(x a)P.5U(x a)P5L(x a)P).(x a)P):=(x(x R x a P(x)x(x R x a P(x)xx R x a P(x)=(x a)P.q(limnan=a):=0,N(n N|an a|0,Nn(n N|an a|).5.1?f(x)ma,b?(limxaf(x)=A)0,0,x R(0|x a|0,0,x,y a,b(|x y|f(x)f(y)|0,0,x,y a,bs.t.(|x y|0).5.2?f(x),fn(x)(n=1,2,3,.)?3ma,b?(0,N,n N,x a,b|fn(x)f(x)|0,N,n N,xn a,b|fn(xn)f(xn)|.(0 M n).6XS(Relations and orderings)6.1 eX,Y 8X Y?f8RlX?Y?/X0 AO/(eY=X,KX2=X X?f8X?X)kxRy5L(x,y)R.6.2 X?XRX?dXee?nv(i)x X,kxRx.(g5)(ii)ex,y X,KxRy yRx.(5)(iii)ex,y,z X,xRy,yRz,KxRz.(D45)k”5LX?dX?5X?dXX?m?X”?dXe6XS(RELATIONS AND ORDERINGS)22(i)x X,kx x.(g5)(ii)ex,y X,Kx y y x.(5)(iii)ex,y,z X,x y,y z,Kx z.(D45)e”X?dXx yL”x?duy”.6.3?X=R1,x y ex y Q,K”R1?dX?AuX2?8R=(x,y)|x,y R1,x y Q,?x y (x,y)R.c?N?T:X Y lX?Y?Xk6.4?X,Y 8lX?Y?XR?e(xRy1)(xRy2)(y1=y2).?X/0 5?X?Y?X?XX=R1=Y,R=0,1 0,1=(x,y)|x 0,1,y 0,1 X?X?(12R12)(12R13);12=13!ef:X Y N?”:=(x,y)X Y|y=f(x):=(x,f(x)|x X.X Y,/?X0 exyyz,K(x,y)x,(x,y)x,y=f(x),z=f(x),y=z.#?/0?VgSN?”/0 e”X?dXxLx3?da=x:=y X|y x.w,X=xXx.6XS(RELATIONS AND ORDERINGS)23eyex y 6=,Kx=y.ex y 6=,Kz x y,?z xz y.w x,w x,x z,?w z,?z y?w y,=w y,?kx y,ny x.ddke?n6.5 e”X?dXKXLX?da?Xua,b Z,m Z+,a b a b=mk,k Z,a b Pa b(modm),X=0 1 .m 1.6.6?M8X=P(M)M?Nf8?8a,b P(M),Ken7k(1)a b.(2)b a.(3)=ab?f8ba?f8M?f8?X5X=P(M)?XaRb:=(a b)dX2?f8?RLR=(a,b)X2|a b.Xw,ven(i)aRa.(reflexivityg5)(ii)(aRb)(bRc)aRc.(transitivityD45)(iii)(aRb)(bRa)a=b.(antisymmetry5)?X?XRevnKX?S6XS(RELATIONS AND ORDERINGS)24X(Partial ordering),da b5LaRb.(X,)S85(X,)S8U?yab X(a 6 b)(b a),X?fP(M),).e(X,)S8va Xb X(a b)(b a)K”X?S(X,)?S8(5S8).X38R1d?”5R1?SK(R1,)?S8e(X,)S8x XX?4?(4?),ey Xx y(y x),Kx=y.5?S8(X,).4?(4?)73(X(R1,)Q4?4?4?=3(X,)?S85X?4?/?0=ky X,y x!e(X,)S8E X,x XE?.(e.)ey E,ky x(x y),E?(e).73Ee(X,)S8E X,K”E?SX?(E,)S8E?4?7E?.(E,)?S8(e(E,)?S8xE?4?ey E,?y xK7kx y,l?x=y,?x xo?gdy E,ky x,=xE?.)e(X,)?S8?f8k?4?K(X,)S8(well ordering),”X?SXg,8N=1,2,.U?”S86XS(RELATIONS AND ORDERINGS)25e8?(dJnK6.7(Hausdorff4?n)z S8k4?Sf8(=e38E X,?(E,)?S83F X,?F?Sf8E$F.)?d?e?K6.8(Zornn)eX S8X?z?Sf8k.KXk4?5yeK6.7?duK6.8.y 6.7 6.8 dK6.7,S8(X,)k4?Sf8E,dEk.x,5yx X?4?duxE?.y E,y x,ekz X,x z?x 6=z,KE?45E z?Sf8x E.lx 6=zE z?S8?w,ez E,Kz x z,Lx=z.(S?)guz 6=x.?x=z,l?xX?4?6.8 6.7?(X,)S8-M=e|eX?Sf8MX SR,e1,e2 M,e1Re2X?f8e1 e2,PR”,(M,)S85?y(M,)vZornn?eN M?Sf8Ke0=eNeN?.dy(i)e0 M(ii)e N,e e0(X?f8)ye0 MLye0X?Sf8x1,x2 e0,3e1 N,e2 N,x1 e1,x2 e2,duN?S8”?e1 e2,?x1,x2 e2,?e2X?Sf8?x16 x2,x26 x17y?e0X?Sf8=e0 M.(ii)e Nw,X?f8e e0,?e0N?.?dZornn(M,)k4?e,d4?e M,?e M,e e e,K e=e.eX?4?Sf8?y.K6.9 SK(The well ordering Principle)z8X,3X?SX”,(X,)S86XS(RELATIONS AND ORDERINGS)26y X?:f8x,(x X)o3x Sa,b x,a 6 b:a=b,Kx S8w,x?S8S8-WXkf8?S?Ndc?W 6=.Ke W,eX X?f8?|X2=X X?X(W,)S8e(W,)?yZornn?N W?Sf8X6.86.7?y?N k.?dZornnWk4?R W,?F W,eR F(X2=XX?f8),KR=F.eyRX?SdW?R W3E X,RE?S(E,R)S8IyE=X=y?x0 XE,K38E x0 SXR,SE?S*?E x0.(i)x E,5x0Rx,(ii)x,y E,5xRy xRy.w,X2?8R R:x,y E,(x,y)RxRy:xRy(x,y)R.(5R E E X X,R(E x0)(E x0).w,RE x0?S?R W,R R,(E x0,R)?S8F E x0,ex0F,KF E,Fk4?ex0 F,x04?dR4?R=R,?lx0 XE,x E,(x0,x)R(E x0)(E x0),?lR E E(x0,x)R?R$R,?gdXE=,=E=X,=RX?Sy.K6.10 JneXAx8K3N?f:A AX,?z A,f()X.y-X=AX,dK6.9,3X?S?lX6=zXk4?Pf(),f:f()?N?f:A X.K6.10?J78?(CARDINAL NUMBER)276.11(Jn?,Q)eXAx?8K38Y AX,?Y dlzXT?Y X:8().ydn6.10,3N?f:A X=AX,?f()X,-Y=f(A),KduXA?Y X=f():8y.yJnHausdorff4?n?oK?d?78?(Cardinal number)X?k8AB?wAB?(#A?=#B)(A?P#A)?AB?31-1A:A B.k8?VgU#A=#B1-1A:A B?2?8kXe7.1 eA,B83lA?B?1-1AN?:A B,KABk?AB?PA B,A=BcardA=cardB.cardAA?A?58g?rN?PAB,cardAk8?2eN=1,2,3,.,E=2,4,6,.,K:N E,(i)=2i,i=1,2,.,1-1A?cardN=cardE,?E$N.8?f8?A:k8?7.2?A,B8eAB?f8?A?uB?PA B(cardA cardB,B A).78?(CARDINAL NUMBER)28eA BAB?KA?uB?PA B,5?8A,3(1)qdux A0,ex D,Kx”(1)?m”.exD,Kx Dc=+k=0Ack,?3k0?x Ack0,?x A0Ak0=(A0A1)(A1A2).(Ak01Ak0),?x”(1)?m”.?(1)(2)aq78?(CARDINAL NUMBER)31dn7.6 :Ak Ak+2(k=0,1,2,.)1-1N?lAk+1 Ak,(k=0,1,2,.):Ak Ak+1 Ak+2 Ak+3(k=0,1,2,.)1-1AN?ePA=D (A0 A1)+k=0(A2k+1 A2k+2)+k=1(A2k A2k+1)A1=D (A2 A3)+k=0(A2k+1 A2k+2)+k=2(A2k A2k+1)p?DidD,A2k+1 A2k+2idA2k+1 A2k+2,k=0,1,.A2k A2k+1 A2k+2 A2k+3,k=0,1,.dn7.7A B.y.5(1),(2)okD=+k=0Ak?UkD 6=,XAn=121n,1,D=+n=1An=12,1 6=.n7.8(n)?8X,X P(X),(X 2X).y yXP(X)?f8?XP(X)?-X=x|x X,KX P(X),N?:X X,x X,-(x)=x,1-1N?y?XP(X)?=3X?P(X)?1-1N?f:X P(X).PB=x X|xf(x)(B kU8).KB P(X),u3xB X,?f(xB)=B.exB B,KdB?xBf(xB)=B,?gexBB,KxB f(xB)=B,?g?X?gdXP(X)?l?X P(X).78?(CARDINAL NUMBER)327.9(8)e8Xg,8N?KX8eX 6NKX8n7.10?8Mf8y lM?e1,dM8e2 M e1,e26=e1,5lM?np?e1,e2,.,en.KdM8?en+1 Me1,e2,.,en,en+1eii=1,2,.,n,?GM?e1,e2,.,en,.-M=e1,e2,.,en,.,KM M.i,-(ei)=i N,M?N?1-1A?M=N,M8dn7.1088?8ok8o8n7.11(1)XY 8KX Y 8(2)eXn(n=1,2,.,m)8Kmn=1Xn8(3)eXn(n=1,2,.)8K+n=1Xn8y(1)X 6 N,Y 6 N,?:X (X)N,:Y (Y)N1-1N?Kw,:X Y N N,(x,y)(x),(y)1-1N?yN2=N.n=2,3,.,N2k?(j,k),Uj+k=n,?(1,1),(2,1),(1,2),(3,1),(2,2),(1,3),(4,1),(3,2),(2,3),(1,4).8N1 n8?33=N2=e1,e2,.,en,.,?N2 N.(2)n,3?fn:N Xn,f:N 1,2,.,m mn=1Xn,(n,)f(n,):=f(n).x mn=1Xn,i,x Xi,lfi:N Xi?n N,s.t.fi(n)=x,?f?l?mn=1Xn6 N 1,2,.,m 6 N N=N,mn=1Xn(3)i,3?fi:N Xi,f:N N+i=1Xi,(n,i)f(n,i):=fi(n)Xi.x i=1Xi,m N,x Xi,n N,s.t.fi(n)=x,=f(n,i)=x,?f?,?+i=1Xi6 N N=N,+i=1Xi5Z=0,1,2,3,.,Q?Nkn?8?8Z=0 1,2,.,n,.1,2,.,n,.,du0,1,2,.,n,.,1,2,.,n,.8?Z8?X8?N 6Z 6 N,dBernsteinnZf:Z2 Q,(m,n)Z,?n 6=0,f(m,n)=mn;?n=0,f(m,0)=0.f?Q?Q8?Q8N1 n8?y“?:8d?IG.Cantor(1845-1918)M?8N1 n8?34/80?w?N?Cantor?8?/80(the naive set theory).?nen1,8d?O?2,8d?3,?5vT5?N?8?exP5P(x)Lx k5P(x).3?gx|P(x)838?ee?8?!?!?!?$?$3xM8?fCj?:1900c3nim?ISIPoincare?4/y3?0,?31?c=I(B.Russell 1872-1970)%?8/?0(Paradox),?F?u:?#?1ng?SNXe?8Bg?8|?8=u?8M,P(M)L5MM,KB:=M|P(M)=M|MM.kCantor?n3B/80 y3BuB?du?8B,oP(B)oP(B)eB B,KUP(B)?BB geBB,KP(B)B B?g?5?J?=nu“?nu“?KpgC?f?foy3?Knu“?fATdX5?XJgC?fopgCgC?f?KATgC?fXJgC?fofpgC?f?KTgC?8N1 n8?35f?)?nu“gC?f?=?nu“gC?f?y?:?3?)?3uCantor3J8/80?Vgvk7?u?E/8?8N0L?8?Ir(E.Zemelo(1871-1953)?nNX8?Vg1908cgJ?8nXx?/80nz?6=?(A.A.Frankel),.(Von.Neumann)/?8nNXnz?8?;?8?x8um?;I?-SN?3?Zemelo-FrankelX?yy#?gQ?nX7L)?N5?KZemelo-FrankelX?N5yd”Poincare/?/?“H?+jk?5?%?SkvkH0”go?5V61?p?2002c e0?Zemelo-FrankelnNXn1(?nAxiom of extensionality)k8AB,A=BABk?=x Ax B.n2(f8n?Axiom schema of subsets)u?8A,?5P,38B,dAkv5P?(=eA8PuA?5KB=x A|P(x)8)dd?/80?3eX8X=x X|x 6=x8dn18X,Y,X=Y,=8?P,?8?f88N1 n8?36dn28A,B/?80A B=x A|xB8AO/eX8A X,KAc=X A8n3(nUnion axiom)u?x8M(?8)38M8M?d?N3M?,?=M:=x|(X M)(x X),eAx8K A8A=x|()(x A).dd/?0M:=x M|X(X M)(x X)A:=x|()(x A).n4(SnPairing axiom)u8X,Y,38Z?Z=kX,Y?(ZPZ:=X,Y XY?S)Z=?X=Y.5X,Y?kS(X,Y):=X,X,X,Y.n5(8nPower set axiom)?8X,38X?Nf8?=P(X):=A|AX?f8.u8XY,d8nnP(P(X)P(Y)8?df8nX Y:=p (P(X)P(Y)|(p=(x,y)(x X)(y Y)8N1 n8?378XY?(k?(Cartesian product)(R.Descartes:1596-1650,I!n!)A?C)5?p=(x,y),(x X,y Y),ATn)(ckS?)x,yw:8(x,y):=(x,y)=x,x,x,y.p=(x,y)?8e1=x,x=xe2=x,y.e1P(X)?f8=e1?x P(X),?P(X)P(Y)?f8e2?x,y P(X)P(Y),?e2P(X)P(Y)?f8?p=(x,y)P(X)P(Y)?f8l?p=(x,y)P(P(X)P(Y).n1-5/#8?dcy?Cantorn?8X,cardX cardX cardP(X)?g?u8?k/?N8?808?Qenku?8X,PX+=XXX?”?U”(Successor).(pXw:8).88B8(inductive),e8?/?U0?n6(nAxiom of infinity)38B8|1-46?Eg,8N?IO?.N0?8B8(?N8B8?)N?,+=,+=,.9N2 dJnyHAUSDORFF4?n38A0,1,2,.n7(JnAxiom of choice Zermlo n)?x838C,?Tx?8X,X CTd?x8F=X,3N?f:F X,?X F,f(X)X.9N2 dJnyHausdorff4?nXJFx8 F,F?/f0(sub chain),eU8?X?S?=A ,B ,oA B,oB A.?N?8?n1b?F8X?f8?8azF?f?EuF,?g lF?F?N?(A F,g(A)F),s.t.A g(A)g(A)A?K3A F,?g(A)=A.y?A0 F,F0 FuA0?even(a)A0 F0(b)zF0?f?uF0(c)eA F0,Kg(A)F0.d?x?-F1=A F|A0 A,KF1(a)A0 A0,A0 F,?A0 F1.(b)F1?feF,9N2 dJnyHAUSDORFF4?n39KB eF,A B,A BFB F,?BFB F1.(c)A F1 F,A g(A)F,?g(A)F1,F1-F0k?KF0(F0AdA0?5duA0=g(A0)A0)?F0?fxvkXJA F0,KA0 A,eyF0F?f-k?C F0?8A F0,A CC A7=C F0|A F0,7kC AA C.C ,-(C)=A F0|A CC A.vc5(a),(b).(a)A0 F0,?A F0,A0 A.(b)?fK,K?X?S8 BKB F0,F0,F0,yy BKB .A F0,lK ,B K,oA B,oB A.?B K,A B,KA B BKB.eB K,B A,K BKB A,?BKB F0,?BKB .C ,(C)v5(a),(b).(a)lC F0A0 F0,F0A0 Co?A0(C).(b)eK0(C)?f BKB F0,KB K,B (C),?B Cg(C)B.e?B KkB C,K BKB C,BKB (C).KB K,g(C)B BKB,?k BKB (C).?C ,b?A (C),yg(A)(C),eA (C)KkenU5(i)A$C,(ii)A=C,(iii)g(C)A.9N2 dJnyHAUSDORFF4?n40(i)eA$C,KCUg(A)?f8Kg(A)A?g?gdC ,g(A)C.(A (C)F0,?g(A)(C).d(c),g(A)F0,?lC og(A)C,oC g(A).eC g(A),KdC$g(A)kC=g(A).?okg(A)C)(ii)eA=C,Kg(A)=g(C),=g(A)g(C)g(A)(C).(iii)eg(C)A,KlA g(A)g(C)g(A)g(A)(C).?g(A)(C)?A (C)l?(C)F0?4?5X?C ,(C)=F0.(C)F0?l(C)kF0(C),?(C)=F0(C ).eA F0,C F,(C)=F0.?oA Cg(C)A,Lg(C).(eg(C)AKg(C).eA ClC g(C),KA g(C)Lg(C).)?F0 =F0.?F0?S?=A,C F0=,lA C F0,A CC A=F0?S?-AF0?dF0v(b),A F0.d(c),g(A)F0,duAF0?4?A g(A)F0,?A g(A)A g(A)=A.u8X?Jf,X?zf8EAuE?f(E)E.Jnuz83JHausdorff45nz S8Pk4?Sf8y?F P?k?Sf8?xP?zd?|9N2 dJnyHAUSDORFF4?n41?f8?S8F?5?S8?S?(e()?S?u1,2,o1 2,o2 1.?a,b ,31,2,a 1,b 2,”?1 2.?2?S?a bb a?S?)?fuP?JXJA F,-A=x Ac:=P A|A x F,g(A)=A A=A f(A)A6=?A g(A).?A=g(A)A=.?A6=#(g(A)A)=#f(A)=1.(f(A)A Ac,dnA,g(A)=A,=3A,A=,?x Ac,A xF,?AF?4?(5F38?e?S?)A F,A?S?