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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
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/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
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/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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dup 73 /I put
dup 77 /M put
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dup 88 /X put
dup 89 /Y put
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readonly def
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currentdict end
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 73 /I put
dup 82 /R put
dup 105 /i put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 2 /multiply put
dup 10 /circlemultiply put
dup 14 /openbullet put
dup 24 /similar put
dup 33 /arrowright put
dup 50 /element put
dup 67 /C put
dup 84 /T put
dup 106 /bar put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 81 /Q put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 25 /germandbls put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 53 /five put
dup 54 /six put
dup 58 /colon put
dup 61 /equal put
dup 65 /A put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 90 /Z put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 122 /z put
dup 127 /dieresis put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 69 /E put
dup 75 /K put
dup 85 /U put
dup 97 /a put
dup 98 /b put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 116 /t put
dup 117 /u put
dup 122 /z put
dup 127 /dieresis put
readonly def
/FontBBox{-33 -250 945 749}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 46 /period put
dup 48 /zero put
dup 50 /two put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 65 /A put
dup 66 /B put
dup 68 /D put
dup 71 /G put
dup 73 /I put
dup 77 /M put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 127 /dieresis put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
currentdict end
currentfile eexec
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cleartomark
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TeXDict begin 39139632 55387786 1000 600 600
(D:/F/Data1/UNI/INSTITUT/VORLESUN/SS00/UEBUNG07.dvi)
@start /Fa 205[47 50[{}1 66.4176 /CMSY8 rf /Fb 174[94
81[{}1 99.6264 /CMEX10 rf /Fc 141[44 4[85 3[33 2[48 12[57
81 5[74 4[94 3[43 7[73 5[27 59[{}11 99.6264 /CMMI12 rf
/Fd 150[24 22[53 8[31 73[{}3 66.4176 /CMMI8 rf /Fe 149[28
21[54 16[52 16[66 16[100 8[77 9[50 3[77 7[77 1[77{}10
99.6264 /CMSY10 rf /Ff 174[77 81[{}1 99.6264 /MSBM10
rf /Fg 128[49 4[43 2[70 51 54 38 38 38 1[54 49 54 81
27 51 30 27 54 49 30 43 54 43 54 49 6[60 3[73 73 70 54
72 1[66 2[89 61 76 1[35 73 77 64 66 75 2[73 3[76 2[27
3[49 49 1[49 49 49 49 1[27 33 27 2[38 38 14[49 25[{}55
99.6264 /CMR12 rf /Fh 128[66 4[58 4[73 51 1[51 2[66 73
4[36 73 66 40 58 73 1[73 66 11[99 9[103 5[90 69[{}18
143.462 /CMR17 rf /Fi 128[56 8[59 62 44 44 46 1[62 56
62 93 31 59 1[31 62 56 34 51 62 50 62 54 10[85 86 78
62 2[77 2[106 3[42 1[88 2[86 1[80 85 8[56 56 56 56 2[56
1[56 1[31 46[{}39 99.6264 /CMBX12 rf end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 600dpi
TeXDict begin
%%PaperSize: A4
 end
%%EndSetup
%%Page: 1 1
TeXDict begin 1 0 bop 115 355 a Fi(Mathematisc)m(hes)39
b(Institut)100 471 y(der)33 b(Univ)m(ersit)742 476 y(\177)743
471 y(at)h(M)984 476 y(\177)981 471 y(unc)m(hen)206 588
y(Prof.)j(Dr.)g(B.)g(P)m(areigis)3104 354 y(SS)h(2000)3124
471 y(Blatt)f(7)413 1084 y Fh(\177)396 1116 y(Ubungen)45
b(zu)f(Einf)1374 1121 y(\177)1370 1116 y(uhrung)g(in)g(die)f
(Kategorien)l(theorie)0 1370 y Fi(Aufgab)s(e)c(25.)i
Fg(In)33 b Fi(V)-9 b(ek)32 b Fg(und)h(in)g Fi(Gr)g Fg(gibt)f(es)h(b)s
(eliebige)h(k)-5 b(ategorielle)33 b(Pro)s(dukte.)0 1663
y Fi(Aufgab)s(e)38 b(26.)j Fg(Zeigen)32 b(Sie:)g(In)g(der)g(Kategorie)g
(aller)f(K)2143 1667 y(\177)2143 1663 y(orp)s(er)h(gibt)g(es)g(im)g
(allgemeinen)h(k)m(eine)0 1779 y(k)-5 b(ategoriellen)33
b(Pro)s(dukte.)h(Es)f(gilt)g(jedo)s(c)m(h)g Ff(Q)23 b
Fe(\002)f Ff(Q)1944 1752 y Fe(\030)1945 1783 y Fg(=)2049
1779 y Ff(Q)p Fg(.)0 2072 y Fi(Aufgab)s(e)39 b(27.)i
Fg(Sei)787 2087 y Fd(R)845 2072 y Fc(M)i Fg(ein)34 b
Fc(R)q Fg(-Linksmo)s(dul.)f(Zeigen)g(Sie:)148 2208 y(\(1\))42
b(F)382 2212 y(\177)379 2208 y(ur)33 b(jede)i(ab)s(elsc)m(he)h(Grupp)s
(e)e Fc(A)g Fg(ist)g(Hom\()p Fc(M)5 b(;)17 b(A)p Fg(\))34
b(ein)h Fc(R)q Fg(-Rec)m(h)m(tsmo)s(dul)g(v)m(erm)3288
2212 y(\177)3288 2208 y(oge)h(der)315 2324 y(Multiplik)-5
b(ation)33 b(\()p Fc(f)11 b(r)s Fg(\)\()p Fc(m)p Fg(\))27
b(:=)h Fc(f)11 b Fg(\()p Fc(r)s(m)p Fg(\).)148 2441 y(\(2\))42
b(Hom\()p Fc(M)5 b(;)17 b Fg(-)o(\))28 b(:)g Fi(Ab)f
Fe(\000)-59 b(!)27 b Fg(Mo)s(d)17 b(-)o Fc(R)34 b Fg(ist)f(ein)g(k)m(o)
m(v)-5 b(arian)m(ter)34 b(F)-8 b(unktor.)148 2557 y(\(3\))42
b(Der)47 b(F)-8 b(unktor)49 b(Hom\()p Fc(M)5 b(;)17 b
Fg(-)o(\))48 b(b)s(esitzt)h(einen)h(linksadjungierten)g(F)-8
b(unktor)48 b(-)32 b Fe(\012)3323 2572 y Fd(R)3414 2557
y Fc(M)65 b Fg(:)315 2673 y(Mo)s(d)16 b(-)p Fc(R)28 b
Fe(\000)-59 b(!)27 b Fi(Ab)p Fg(.)0 2966 y Fi(Aufgab)s(e)55
b(28.)49 b Fg(Sei)e Fe(C)58 b Fg(=)52 b Fi(T)-9 b(op)46
b Fg(die)i(Kategorie)e(der)h(top)s(ologisc)m(hen)h(R)2802
2970 y(\177)2802 2966 y(aume)f(mit)g(stetigen)0 3082
y(Abbildungen)34 b(als)f(Morphismen.)148 3218 y(\(1\))42
b(Sei)32 b(\()p Fc(X)590 3233 y Fd(i)618 3218 y Fe(j)p
Fc(i)c Fe(2)g Fc(I)8 b Fg(\))31 b(eine)i(F)-8 b(amilie)32
b(v)m(on)g(top)s(ologisc)m(hen)h(R)2325 3222 y(\177)2325
3218 y(aumen,)g(sei)f Fc(X)40 b Fg(eine)33 b(Menge)f(und)315
3335 y(seien)37 b Fc(f)605 3350 y Fd(i)668 3335 y Fg(:)e
Fc(X)42 b Fe(\000)-60 b(!)34 b Fc(X)1085 3350 y Fd(i)1150
3335 y Fg(Abbildungen.)k(Zeigen)f(Sie:)g(es)g(gibt)g(genau)g(eine)g(T)
-8 b(op)s(ologie)36 b Fe(T)315 3451 y Fg(auf)c Fc(X)8
b Fg(,)32 b(so)347 3567 y(\(a\))41 b(da\031)32 b(die)i(Abbildungen)g
Fc(f)1480 3582 y Fd(i)1536 3567 y Fg(:)27 b Fc(X)36 b
Fe(\000)-60 b(!)28 b Fc(X)1933 3582 y Fd(i)1993 3567
y Fg(stetig)33 b(sind)h(und)342 3683 y(\(b\))41 b(da\031)f(f)738
3687 y(\177)735 3683 y(ur)h(jeden)g(top)s(ologisc)m(hen)h(Raum)e
Fc(Y)62 b Fg(und)41 b(jede)g(Abbildung)g Fc(f)52 b Fg(:)42
b Fc(Y)62 b Fe(\000)-60 b(!)41 b Fc(X)513 3800 y Fg(diese)34
b(Abbildung)g Fc(f)43 b Fg(genau)33 b(dann)g(stetig)g(ist,)g(w)m(enn)h
(alle)f Fc(f)2757 3815 y Fd(i)2807 3800 y Fe(\016)22
b Fc(f)43 b Fg(stetig)33 b(sind.)148 3916 y(\(2\))42
b(In)i Fi(T)-9 b(op)43 b Fg(gibt)h(es)h(zu)f(jeder)g(F)-8
b(amilie)44 b(v)m(on)h(top)s(ologisc)m(hen)g(R)2650 3920
y(\177)2650 3916 y(aumen)f(\()p Fc(X)3094 3931 y Fd(i)3122
3916 y Fe(j)p Fc(i)j Fe(2)g Fc(I)8 b Fg(\))43 b(ein)315
4032 y(k)-5 b(ategorielles)33 b(Pro)s(dukt)1251 3957
y Fb(Q)1345 4061 y Fd(i)p Fa(2)p Fd(I)1473 4032 y Fc(X)1554
4047 y Fd(i)1582 4032 y Fg(.)100 5563 y(Abgab)s(e:)f(Dienstag,)h(den)g
(20.06.2000)e(11:15)h(Uhr)h(in)g(der)g(V)-8 b(orlesung)p
eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF