!"#$ % !"#$#%&'(&' )*" +(+&," -*.*&*/0&'( !" #$%&'() *+,+)+-# !"!" ! #$%&'()'($ !"#$ %&'&( "#$ %$&"#$' ($&) &$)"* (+,+#+-")"'"' ./01 %$ 1.,.#0 12*$#$&"'"' )0&.)03#.4 S '"' ./01 %$ 1.,.#0 12*$#$&"'"' ($&) &$)"*#$&"'5$' "6.&$( +#5787'7 9:)($&"'";! !"#$% f : X → (Y, S ) <+'1)"3+'7'7' X 2;$&"'$ 1+'57&5787 ";52=$# (+,+#+-"3$ >($&) &$)"* (+,+#+-")"? H 5"3$#"*! @'$&*$ A! ! 73.&0'B. S = {f −1 (S) : S ∈ S } >A! ? ."#$)" H (+,+#+-")"'"' 6"& .#(C(.6.'050&! D7 ."#$'"' )+'#7 .&.1$)"(#$&"'"' +#7=(7&5787 ."#$3" S "#$ 9:)($&$#"*! ! n n \ \ −1 f Si = f −1 (Si ) i=1 f −1 [ ı∈I i=1 Sı ! = [ f −1 (Sı ) ı∈I $="(#"1#$&"'5$' S = S = S = H +#5787 9:&2#2&! E F.#5$4 A! 9$&$8"'B$ ∗ T ∈ H ⇔ (∃S ∈ S )(T = f −1 (S) GH !"#$ %& '!(#)#$"*+"* ,-('.+."/( 0-1-"-23"*+ !"#$ %&'#(' $)$*+,- ./0#1&,02 L 34('5" S 678797:"5"&!' 308091 "5' f −1 (L) 6'#5 #'5(" H 678797:"5"&!' 308091 790,0361#$ ;0#<16 790#032 K 34('5" H 678797:"5"&!' 308091 "5' K ′ 0=1361#$ >$* ?'#'@"&,'2 K ′ = f −1 (S) 790,03 A"="(!' A"# S ∈ S 0=13 34('5" B0#!1#$ S ′ 308091!1# B' K ′ = f −1 (S ′ ) !4#$ C D09!'2 H 678797:"5"&"& 308091 34('9'#" S 678797:"5"&"& 308091 34('9'#"&"& 6'#5 #'5"(9'#"!"#$ )$ E"&' /.30#1!03" F#&'36'2 D'# x ∈ X "="& W 34('9'#" S /' ?F#' f (x) &736051&1& A"# 37(<.9.390# 60A0&1&1 60#1/7#50 X 4G'#"&!' S &"& 6'#5 #'5"( 678797:"5"&' ?F#'2 f −1 (W ) 34('9'#" x &736051&1& A"# 37(<.9.390# 60A0&1&1 60#0/0,0361#$ HF56'#"&"G$ !"#$% W ∈ B(f (x)) ⇒ (∃S ∈ S )(f (x) ∈ S ⊂ W ⇒ (x ∈ f −1 (S) ⊂ f −1 (W ) ∈ B((f (x)) B ∈ B(x) ⇒ (∃T ∈ H )(x ∈ T ⊂ B) ⇒ (∃S ∈ S )(x ∈ T = f −1 (S) ⊂ B) I$ (X, T B' (Y, S ) .G0/90#1 "9' A"# f : X → Y J7&35"/7&. B'#"95"&$ f &"& 54#'39" 79(051 "="& ?'#'39" B' /'6'#9" 37<.92 X 4G'#"&!' S &"& 6'#5 #'5"( 678797:"5" 790& H 678797:"5"&"& T 678797:"5"&!'& !0D0 30A0 !73.9. 79K (051!1#$ HF56'#"&"G$ !"#$% f : (X, T ) → (Y, S ) J7&35"/7&. 54#'39" "5' S ∈ S ⇒ f −1 (S) ∈ T !"#$ L.#0!0&2 >$* ?'#'@"&,'2 S∗ = H ⊂ T 79.#$ M$ !"#$ %&'&( ?F56'#"(9'#" 0961&!02 H "G!4<'9 678797:"5"2 X 4G'#"&!'3" 678797K :"9'# 0#051&!0 D'# ı ∈ I "="& fı FG!'<9"3 J7&35"/7&.&. 54#'39" 3190& 678797:"K 9'#"& 0#03'5"6"&' '<"66"#$ HF56'#"&"G$ L.&.& "#!)# *&+&, "9' '<!'@'# 79!.@.&. ?F56'#"&"G$ !"#$% L464& fı : X → (X, Tı ) 54#'39" FG!'<9"3 J7&35"/7&90#1&1& 0"9'5"&" F = {fı } "9' B' 5FG37&.5. 678797:"9'#"& 0#03'5"6"&" \ T = Tı +>$)ı∈I "9' ?F56'#'9"($ L0<30 A"# !'/"<9' Tı ∈ T ⇔ fı : X → (X, Tı ) 54#'39" A0@1&6151 B0#!1#$ H "G!4<'9 678797:"5" F /' 0"6 D'# fı J7&35"/7&.&. 54#'39" 3190#$ C D09!'2 >$) ?'#'@"&,' H ⊃ T !"#$ ;0#<16 790#032 H &1& 60&1(1 ./0#1&,02 T ∈ T ⇒ (∀ı ∈ I)(T = fı−1 (T ) ∈ H !"! #$%&'() *+,+)+-# ! "#$% &'()'*) T = H +(,$% &-) './"./0 &/)$1) 2%3%3 4)$)56/7) S = {fı−1 (T ) : T ∈ Tı } .6()*6 H = T 68"9:)( -+;+(+<6*6/6/ =6$ .(->-.=./#"#$% &/)$1) !%?%@ ,'.$#/7.0 S .6()*6/6 .(- -.=./ A.=,( )")/ H -+;+(+<6*6 Tı -+;+(+<6()$6/6/ )/ A9B9A 9*- *#/#$#"#$% C% D6$ X A91)*6 6() =6$ Y = {(Y ı, Tı ) : ı ∈ I} -+;+(+<6A ,8.'(.$ .6()*6 E)$6(6'+$% F5)$ F = {fı : X → Y ı} *.=6- G+/A*6'+/(.$"./ +(,:./ =6$ .6() 6*)0 F ') 4H$) Y /6/ X 98)$6/")A6 68"9:)( -+;+(+<6*6/6 =,(,/,8% !"#$% X 98)$6/")A6 .'$#A +(1.'./ -+;+(+<6 F .6()*6/) .6- =9-9/ G+/A*6'> +/(.$# *9$)A(6 A#(.$ I=A8%J$+=()1 %?%3I@KK%L M.(")0 H/7)A6 ;$+=()1 4)$)56/7)0 .'$#A -+;+(+<6 .$././ 68"9:)( -+;+(+<6"6$% % R 4)$B)( *.'#(.$ A91)*6 98)$6/") F = {fab : R → (R, R) | fab : x ax + b, a, b ∈ R} G+/A*6'+/(.$#/#0 './6 R ")/ R ') =9-9/ "+5$,*.( G+/A*6'+/(.$# *9$)A(6 A#(./ )/ A.=. "+A,(, -+;+(+<6/6/ '6/) 4)$B)( )A*)/6/ R *.(- -+;+(+<6*6 +(",5,/, 4H*-)$6/68% !"#$% N)$ fab ∈ F "+5$,*.( "H/9:919/9/ =6$ -+;+(+<6A ):'.;# "H/9:919 +(",5,/, =6(6'+$,8 I=A8 &$/)A %O%@K% D,$."./ −1 T ∈ R ⇔ fab (T ) ∈ R −1 +(",5, 4H$9(9$% L M.(")0 M)$ a, b ∈ R 6B6/ fab (R) = R +(.7.A-#$0 A6 =, −1 {fab (R) : a, b ∈ R} = R ):6-(656/6/ E.$(#5# ")1)A-6$% D, ):6-(656/ *+( './#0 F .6()*6/6/ R 98)$6/) A+/",$",5, 68"9:)( -+;+(+<6"6$% !% R 98)$6/")A6 .(->(616- -+;+(+<6*6/6 L 6() 4H*-)$)(61% R 98)$6/") F = {fab : R → (R, L ) | fab : x ax + b, a, b ∈ R} "+5$,*.( "H/9:91()$6/6 *9$)A(6 A#(./ )/ A.=. -+;+(+<6/6/ .'$#A -+;+(+<6> ")/ =.:A.*# +(.1.'.7.5#/#P './60 (R, T ) ,8.'#/#/ F ') 4H$) R 98)$6/")A6 68"9:)( -+;+(+<6/6/ .'$#A -+;+(+<6 +(",5,/, 4H*-)$6/68% ! !"#$ %& '!(#)#$"*+"* ,-('.+."/( 0-1-"-23"*+ !"#$% "#$%&'()( *$+,-./ 0&1&/&2*3* T */. 4#)0.5./*67 8.5 p ∈ R *9*' {p} %,6.)*'*' T :;9<% &/+(=('( 4#)0.56./*3*$7 f (x) = x − p >. g(x) = −x + p ?&'%)*3&'/;5< F ;*/.)*'. ;*00*5/.57 B = [0, p) %,6.)* *). L 0&1&/&: 2*)*'. ;*00*57 f −1 (B) = [p, 2p), g −1 (B) = (0, p] 0.5) 5.)*6/.5* T *$+,-./ 0&1&/&2*)*'*' ;/0:0;@;'<'; ;*00*57 A B;/+. @('/;5<' ;5;%.)*0* [p, 2p) ∩ (0, p] = {p} %,6.)* T 0&1&/&2*)*'. ;*0 &/;C;%0<5D 3;'* B.5 p ∈ R *9*' {p} ∈ T +*57