:: T_1TOPSP semantic presentation begin theorem Th1: :: T_1TOPSP:1 for T being non empty TopSpace for A being non empty a_partition of the carrier of T for y being Subset of (space A) holds (Proj A) " y = union y proof let T be non empty TopSpace; ::_thesis: for A being non empty a_partition of the carrier of T for y being Subset of (space A) holds (Proj A) " y = union y let A be non empty a_partition of the carrier of T; ::_thesis: for y being Subset of (space A) holds (Proj A) " y = union y let y be Subset of (space A); ::_thesis: (Proj A) " y = union y reconsider y = y as Subset of A by BORSUK_1:def_7; (Proj A) " y = (proj A) " y by BORSUK_1:def_8 .= union y by EQREL_1:67 ; hence (Proj A) " y = union y ; ::_thesis: verum end; theorem Th2: :: T_1TOPSP:2 for T being non empty TopSpace for S being non empty a_partition of the carrier of T for A being Subset of (space S) for B being Subset of T st B = union A holds ( A is closed iff B is closed ) proof let T be non empty TopSpace; ::_thesis: for S being non empty a_partition of the carrier of T for A being Subset of (space S) for B being Subset of T st B = union A holds ( A is closed iff B is closed ) let S be non empty a_partition of the carrier of T; ::_thesis: for A being Subset of (space S) for B being Subset of T st B = union A holds ( A is closed iff B is closed ) let A be Subset of (space S); ::_thesis: for B being Subset of T st B = union A holds ( A is closed iff B is closed ) let B be Subset of T; ::_thesis: ( B = union A implies ( A is closed iff B is closed ) ) reconsider C = A as Subset of S by BORSUK_1:def_7; A1: ([#] T) \ (union A) = (union S) \ (union C) by EQREL_1:def_4 .= union (S \ A) by EQREL_1:43 .= union (([#] (space S)) \ A) by BORSUK_1:def_7 ; assume A2: B = union A ; ::_thesis: ( A is closed iff B is closed ) thus ( A is closed implies B is closed ) ::_thesis: ( B is closed implies A is closed ) proof reconsider om = ([#] (space S)) \ A as Subset of S by BORSUK_1:def_7; assume A is closed ; ::_thesis: B is closed then ([#] (space S)) \ A is open by PRE_TOPC:def_3; then om in the topology of (space S) by PRE_TOPC:def_2; then ([#] T) \ B in the topology of T by A2, A1, BORSUK_1:27; then ([#] T) \ B is open by PRE_TOPC:def_2; hence B is closed by PRE_TOPC:def_3; ::_thesis: verum end; thus ( B is closed implies A is closed ) ::_thesis: verum proof reconsider om = ([#] (space S)) \ A as Subset of S by BORSUK_1:def_7; assume B is closed ; ::_thesis: A is closed then ([#] T) \ B is open by PRE_TOPC:def_3; then ([#] T) \ (union A) in the topology of T by A2, PRE_TOPC:def_2; then om in the topology of (space S) by A1, BORSUK_1:27; then ([#] (space S)) \ A is open by PRE_TOPC:def_2; hence A is closed by PRE_TOPC:def_3; ::_thesis: verum end; end; theorem Th3: :: T_1TOPSP:3 for T being non empty TopSpace holds { A where A is a_partition of the carrier of T : A is closed } is Part-Family of the carrier of T proof let T be non empty TopSpace; ::_thesis: { A where A is a_partition of the carrier of T : A is closed } is Part-Family of the carrier of T set S = { A where A is a_partition of the carrier of T : A is closed } ; A1: now__::_thesis:_for_B_being_set_st_B_in__{__A_where_A_is_a_partition_of_the_carrier_of_T_:_A_is_closed__}__holds_ B_is_a_partition_of_the_carrier_of_T let B be set ; ::_thesis: ( B in { A where A is a_partition of the carrier of T : A is closed } implies B is a_partition of the carrier of T ) assume B in { A where A is a_partition of the carrier of T : A is closed } ; ::_thesis: B is a_partition of the carrier of T then ex A being a_partition of the carrier of T st ( B = A & A is closed ) ; hence B is a_partition of the carrier of T ; ::_thesis: verum end; { A where A is a_partition of the carrier of T : A is closed } c= bool (bool the carrier of T) proof let B be set ; :: according to TARSKI:def_3 ::_thesis: ( not B in { A where A is a_partition of the carrier of T : A is closed } or B in bool (bool the carrier of T) ) assume B in { A where A is a_partition of the carrier of T : A is closed } ; ::_thesis: B in bool (bool the carrier of T) then ex A being a_partition of the carrier of T st ( B = A & A is closed ) ; hence B in bool (bool the carrier of T) ; ::_thesis: verum end; hence { A where A is a_partition of the carrier of T : A is closed } is Part-Family of the carrier of T by A1, EQREL_1:def_7; ::_thesis: verum end; definition let T be non empty TopSpace; func Closed_Partitions T -> non empty Part-Family of the carrier of T equals :: T_1TOPSP:def 1 { A where A is a_partition of the carrier of T : A is closed } ; coherence { A where A is a_partition of the carrier of T : A is closed } is non empty Part-Family of the carrier of T proof reconsider ct = { the carrier of T} as a_partition of the carrier of T by EQREL_1:39; set F = { A where A is a_partition of the carrier of T : A is closed } ; for A being Subset of T st A in ct holds A is closed proof let A be Subset of T; ::_thesis: ( A in ct implies A is closed ) assume A in ct ; ::_thesis: A is closed then A = [#] T by TARSKI:def_1; hence A is closed ; ::_thesis: verum end; then ct is closed by TOPS_2:def_2; then ct in { A where A is a_partition of the carrier of T : A is closed } ; hence { A where A is a_partition of the carrier of T : A is closed } is non empty Part-Family of the carrier of T by Th3; ::_thesis: verum end; end; :: deftheorem defines Closed_Partitions T_1TOPSP:def_1_:_ for T being non empty TopSpace holds Closed_Partitions T = { A where A is a_partition of the carrier of T : A is closed } ; definition let T be non empty TopSpace; func T_1-reflex T -> TopSpace equals :: T_1TOPSP:def 2 space (Intersection (Closed_Partitions T)); correctness coherence space (Intersection (Closed_Partitions T)) is TopSpace; ; end; :: deftheorem defines T_1-reflex T_1TOPSP:def_2_:_ for T being non empty TopSpace holds T_1-reflex T = space (Intersection (Closed_Partitions T)); registration let T be non empty TopSpace; cluster T_1-reflex T -> non empty strict ; coherence ( T_1-reflex T is strict & not T_1-reflex T is empty ) ; end; theorem Th4: :: T_1TOPSP:4 for T being non empty TopSpace holds T_1-reflex T is T_1 proof let T be non empty TopSpace; ::_thesis: T_1-reflex T is T_1 now__::_thesis:_for_p_being_Point_of_(T_1-reflex_T)_holds_{p}_is_closed let p be Point of (T_1-reflex T); ::_thesis: {p} is closed reconsider I = (Intersection (Closed_Partitions T)) \ {p} as Subset of (Intersection (Closed_Partitions T)) by XBOOLE_1:36; A1: the carrier of (T_1-reflex T) = Intersection (Closed_Partitions T) by BORSUK_1:def_7; then consider x being Element of T such that A2: p = EqClass (x,(Intersection (Closed_Partitions T))) by EQREL_1:42; reconsider q = p as Subset of T by A2; A3: { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } c= bool the carrier of T proof let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } or Z in bool the carrier of T ) assume Z in { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } ; ::_thesis: Z in bool the carrier of T then ex Y being a_partition of the carrier of T st ( Z = EqClass (x,Y) & Y in Closed_Partitions T ) ; hence Z in bool the carrier of T ; ::_thesis: verum end; not { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } is empty proof consider Y being set such that A4: Y in Closed_Partitions T by XBOOLE_0:def_1; reconsider Y = Y as a_partition of the carrier of T by A4, EQREL_1:def_7; EqClass (x,Y) in { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } by A4; hence not { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } is empty ; ::_thesis: verum end; then reconsider m = { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } as non empty Subset-Family of T by A3; reconsider m = m as non empty Subset-Family of T ; A5: for A being Subset of T st A in m holds A is closed proof let A be Subset of T; ::_thesis: ( A in m implies A is closed ) assume A in m ; ::_thesis: A is closed then consider S being a_partition of the carrier of T such that A6: ( A = EqClass (x,S) & S in Closed_Partitions T ) ; ( ex B being a_partition of the carrier of T st ( S = B & B is closed ) & A in S ) by A6, EQREL_1:def_6; hence A is closed by TOPS_2:def_2; ::_thesis: verum end; p = meet { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } by A2, EQREL_1:def_8; then q is closed by A5, PRE_TOPC:14; then ([#] T) \ q is open by PRE_TOPC:def_3; then A7: ([#] T) \ p in the topology of T by PRE_TOPC:def_2; p in Intersection (Closed_Partitions T) by A1; then union ((Intersection (Closed_Partitions T)) \ {p}) in the topology of T by A7, EQREL_1:44; then A8: I in { A where A is Subset of (Intersection (Closed_Partitions T)) : union A in the topology of T } ; reconsider I = I as Subset of (space (Intersection (Closed_Partitions T))) by BORSUK_1:def_7; reconsider I = I as Subset of (T_1-reflex T) ; ( the topology of (space (Intersection (Closed_Partitions T))) = { A where A is Subset of (Intersection (Closed_Partitions T)) : union A in the topology of T } & I = ([#] (T_1-reflex T)) \ {p} ) by BORSUK_1:def_7; then ([#] (T_1-reflex T)) \ {p} is open by A8, PRE_TOPC:def_2; hence {p} is closed by PRE_TOPC:def_3; ::_thesis: verum end; hence T_1-reflex T is T_1 by URYSOHN1:19; ::_thesis: verum end; registration let T be non empty TopSpace; cluster T_1-reflex T -> T_1 ; coherence T_1-reflex T is T_1 by Th4; end; registration cluster non empty TopSpace-like T_1 for TopStruct ; existence ex b1 being TopSpace st ( b1 is T_1 & not b1 is empty ) proof set T = the non empty TopSpace; take T_1-reflex the non empty TopSpace ; ::_thesis: ( T_1-reflex the non empty TopSpace is T_1 & not T_1-reflex the non empty TopSpace is empty ) thus ( T_1-reflex the non empty TopSpace is T_1 & not T_1-reflex the non empty TopSpace is empty ) ; ::_thesis: verum end; end; definition let T be non empty TopSpace; func T_1-reflect T -> continuous Function of T,(T_1-reflex T) equals :: T_1TOPSP:def 3 Proj (Intersection (Closed_Partitions T)); correctness coherence Proj (Intersection (Closed_Partitions T)) is continuous Function of T,(T_1-reflex T); ; end; :: deftheorem defines T_1-reflect T_1TOPSP:def_3_:_ for T being non empty TopSpace holds T_1-reflect T = Proj (Intersection (Closed_Partitions T)); theorem Th5: :: T_1TOPSP:5 for T, T1 being non empty TopSpace for f being continuous Function of T,T1 st T1 is T_1 holds ( { (f " {z}) where z is Element of T1 : z in rng f } is a_partition of the carrier of T & ( for A being Subset of T st A in { (f " {z}) where z is Element of T1 : z in rng f } holds A is closed ) ) proof let T, T1 be non empty TopSpace; ::_thesis: for f being continuous Function of T,T1 st T1 is T_1 holds ( { (f " {z}) where z is Element of T1 : z in rng f } is a_partition of the carrier of T & ( for A being Subset of T st A in { (f " {z}) where z is Element of T1 : z in rng f } holds A is closed ) ) let f be continuous Function of T,T1; ::_thesis: ( T1 is T_1 implies ( { (f " {z}) where z is Element of T1 : z in rng f } is a_partition of the carrier of T & ( for A being Subset of T st A in { (f " {z}) where z is Element of T1 : z in rng f } holds A is closed ) ) ) assume A1: T1 is T_1 ; ::_thesis: ( { (f " {z}) where z is Element of T1 : z in rng f } is a_partition of the carrier of T & ( for A being Subset of T st A in { (f " {z}) where z is Element of T1 : z in rng f } holds A is closed ) ) A2: dom f = the carrier of T by FUNCT_2:def_1; thus { (f " {z}) where z is Element of T1 : z in rng f } is a_partition of the carrier of T ::_thesis: for A being Subset of T st A in { (f " {z}) where z is Element of T1 : z in rng f } holds A is closed proof { (f " {z}) where z is Element of T1 : z in rng f } c= bool the carrier of T proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { (f " {z}) where z is Element of T1 : z in rng f } or y in bool the carrier of T ) assume y in { (f " {z}) where z is Element of T1 : z in rng f } ; ::_thesis: y in bool the carrier of T then ex z being Element of T1 st ( y = f " {z} & z in rng f ) ; hence y in bool the carrier of T ; ::_thesis: verum end; then reconsider fz = { (f " {z}) where z is Element of T1 : z in rng f } as Subset-Family of T ; reconsider fz = fz as Subset-Family of T ; A3: for A being Subset of T st A in fz holds ( A <> {} & ( for B being Subset of T holds ( not B in fz or A = B or A misses B ) ) ) proof let A be Subset of T; ::_thesis: ( A in fz implies ( A <> {} & ( for B being Subset of T holds ( not B in fz or A = B or A misses B ) ) ) ) assume A in fz ; ::_thesis: ( A <> {} & ( for B being Subset of T holds ( not B in fz or A = B or A misses B ) ) ) then consider z being Element of T1 such that A4: A = f " {z} and A5: z in rng f ; consider y being set such that A6: ( y in dom f & z = f . y ) by A5, FUNCT_1:def_3; f . y in {(f . y)} by TARSKI:def_1; hence A <> {} by A4, A6, FUNCT_1:def_7; ::_thesis: for B being Subset of T holds ( not B in fz or A = B or A misses B ) let B be Subset of T; ::_thesis: ( not B in fz or A = B or A misses B ) assume B in fz ; ::_thesis: ( A = B or A misses B ) then consider w being Element of T1 such that A7: B = f " {w} and w in rng f ; now__::_thesis:_(_not_A_misses_B_implies_A_=_B_) assume not A misses B ; ::_thesis: A = B then consider v being set such that A8: v in A and A9: v in B by XBOOLE_0:3; f . v in {z} by A4, A8, FUNCT_1:def_7; then A10: f . v = z by TARSKI:def_1; f . v in {w} by A7, A9, FUNCT_1:def_7; hence A = B by A4, A7, A10, TARSKI:def_1; ::_thesis: verum end; hence ( A = B or A misses B ) ; ::_thesis: verum end; the carrier of T c= union fz proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of T or y in union fz ) consider z being set such that A11: z = f . y ; assume A12: y in the carrier of T ; ::_thesis: y in union fz then A13: z in rng f by A2, A11, FUNCT_1:def_3; then reconsider z = z as Element of T1 ; A14: f " {z} in fz by A13; f . y in {(f . y)} by TARSKI:def_1; then y in f " {z} by A2, A12, A11, FUNCT_1:def_7; hence y in union fz by A14, TARSKI:def_4; ::_thesis: verum end; then union fz = the carrier of T by XBOOLE_0:def_10; hence { (f " {z}) where z is Element of T1 : z in rng f } is a_partition of the carrier of T by A3, EQREL_1:def_4; ::_thesis: verum end; thus for A being Subset of T st A in { (f " {z}) where z is Element of T1 : z in rng f } holds A is closed ::_thesis: verum proof let A be Subset of T; ::_thesis: ( A in { (f " {z}) where z is Element of T1 : z in rng f } implies A is closed ) assume A in { (f " {z}) where z is Element of T1 : z in rng f } ; ::_thesis: A is closed then consider z being Element of T1 such that A15: A = f " {z} and z in rng f ; {z} is closed by A1, URYSOHN1:19; hence A is closed by A15, PRE_TOPC:def_6; ::_thesis: verum end; end; theorem Th6: :: T_1TOPSP:6 for T, T1 being non empty TopSpace for f being continuous Function of T,T1 st T1 is T_1 holds for w being set for x being Element of T st w = EqClass (x,(Intersection (Closed_Partitions T))) holds w c= f " {(f . x)} proof let T, T1 be non empty TopSpace; ::_thesis: for f being continuous Function of T,T1 st T1 is T_1 holds for w being set for x being Element of T st w = EqClass (x,(Intersection (Closed_Partitions T))) holds w c= f " {(f . x)} let f be continuous Function of T,T1; ::_thesis: ( T1 is T_1 implies for w being set for x being Element of T st w = EqClass (x,(Intersection (Closed_Partitions T))) holds w c= f " {(f . x)} ) assume A1: T1 is T_1 ; ::_thesis: for w being set for x being Element of T st w = EqClass (x,(Intersection (Closed_Partitions T))) holds w c= f " {(f . x)} then reconsider fz = { (f " {z}) where z is Element of T1 : z in rng f } as a_partition of the carrier of T by Th5; let w be set ; ::_thesis: for x being Element of T st w = EqClass (x,(Intersection (Closed_Partitions T))) holds w c= f " {(f . x)} let x be Element of T; ::_thesis: ( w = EqClass (x,(Intersection (Closed_Partitions T))) implies w c= f " {(f . x)} ) for A being Subset of T st A in fz holds A is closed by A1, Th5; then fz is closed by TOPS_2:def_2; then fz in { B where B is a_partition of the carrier of T : B is closed } ; then A2: EqClass (x,fz) in { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } ; assume A3: w = EqClass (x,(Intersection (Closed_Partitions T))) ; ::_thesis: w c= f " {(f . x)} A4: dom f = the carrier of T by FUNCT_2:def_1; A5: f " {(f . x)} = EqClass (x,fz) proof reconsider fx = f . x as Element of T1 ; f . x in rng f by A4, FUNCT_1:def_3; then A6: f " {fx} in fz ; f . x in {(f . x)} by TARSKI:def_1; then x in f " {(f . x)} by A4, FUNCT_1:def_7; hence f " {(f . x)} = EqClass (x,fz) by A6, EQREL_1:def_6; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in w or y in f " {(f . x)} ) A7: EqClass (x,(Intersection (Closed_Partitions T))) = meet { (EqClass (x,S)) where S is a_partition of the carrier of T : S in Closed_Partitions T } by EQREL_1:def_8; assume y in w ; ::_thesis: y in f " {(f . x)} hence y in f " {(f . x)} by A3, A2, A5, A7, SETFAM_1:def_1; ::_thesis: verum end; theorem Th7: :: T_1TOPSP:7 for T, T1 being non empty TopSpace for f being continuous Function of T,T1 st T1 is T_1 holds for w being set st w in the carrier of (T_1-reflex T) holds ex z being Element of T1 st ( z in rng f & w c= f " {z} ) proof let T, T1 be non empty TopSpace; ::_thesis: for f being continuous Function of T,T1 st T1 is T_1 holds for w being set st w in the carrier of (T_1-reflex T) holds ex z being Element of T1 st ( z in rng f & w c= f " {z} ) let f be continuous Function of T,T1; ::_thesis: ( T1 is T_1 implies for w being set st w in the carrier of (T_1-reflex T) holds ex z being Element of T1 st ( z in rng f & w c= f " {z} ) ) assume A1: T1 is T_1 ; ::_thesis: for w being set st w in the carrier of (T_1-reflex T) holds ex z being Element of T1 st ( z in rng f & w c= f " {z} ) let w be set ; ::_thesis: ( w in the carrier of (T_1-reflex T) implies ex z being Element of T1 st ( z in rng f & w c= f " {z} ) ) assume w in the carrier of (T_1-reflex T) ; ::_thesis: ex z being Element of T1 st ( z in rng f & w c= f " {z} ) then w in Intersection (Closed_Partitions T) by BORSUK_1:def_7; then consider x being Element of T such that A2: w = EqClass (x,(Intersection (Closed_Partitions T))) by EQREL_1:42; reconsider x = x as Element of T ; reconsider fx = f . x as Element of T1 ; take fx ; ::_thesis: ( fx in rng f & w c= f " {fx} ) dom f = the carrier of T by FUNCT_2:def_1; hence ( fx in rng f & w c= f " {fx} ) by A1, A2, Th6, FUNCT_1:def_3; ::_thesis: verum end; theorem Th8: :: T_1TOPSP:8 for T, T1 being non empty TopSpace for f being continuous Function of T,T1 st T1 is T_1 holds ex h being continuous Function of (T_1-reflex T),T1 st f = h * (T_1-reflect T) proof let T, T1 be non empty TopSpace; ::_thesis: for f being continuous Function of T,T1 st T1 is T_1 holds ex h being continuous Function of (T_1-reflex T),T1 st f = h * (T_1-reflect T) let f be continuous Function of T,T1; ::_thesis: ( T1 is T_1 implies ex h being continuous Function of (T_1-reflex T),T1 st f = h * (T_1-reflect T) ) set g = T_1-reflect T; A1: dom (T_1-reflect T) = the carrier of T by FUNCT_2:def_1; defpred S1[ set , set ] means for z being Element of T1 st z in rng f & \$1 c= f " {z} holds \$2 = f " {z}; assume A2: T1 is T_1 ; ::_thesis: ex h being continuous Function of (T_1-reflex T),T1 st f = h * (T_1-reflect T) then reconsider fx = { (f " {x}) where x is Element of T1 : x in rng f } as a_partition of the carrier of T by Th5; A3: dom f = the carrier of T by FUNCT_2:def_1; A4: for y being set st y in the carrier of (T_1-reflex T) holds ex w being set st S1[y,w] proof let y be set ; ::_thesis: ( y in the carrier of (T_1-reflex T) implies ex w being set st S1[y,w] ) assume y in the carrier of (T_1-reflex T) ; ::_thesis: ex w being set st S1[y,w] then y in Intersection (Closed_Partitions T) by BORSUK_1:def_7; then consider x being Element of T such that A5: y = EqClass (x,(Intersection (Closed_Partitions T))) by EQREL_1:42; reconsider x = x as Element of T ; set w = f " {(f . x)}; take f " {(f . x)} ; ::_thesis: S1[y,f " {(f . x)}] let z be Element of T1; ::_thesis: ( z in rng f & y c= f " {z} implies f " {(f . x)} = f " {z} ) assume that A6: z in rng f and A7: y c= f " {z} ; ::_thesis: f " {(f . x)} = f " {z} reconsider fix = f . x as Element of T1 ; f . x in rng f by A3, FUNCT_1:def_3; then A8: f " {fix} in fx ; not y is empty by A5, EQREL_1:def_6; then A9: ex z1 being set st z1 in y by XBOOLE_0:def_1; f " {z} in fx by A6; then A10: ( f " {(f . x)} misses f " {z} or f " {(f . x)} = f " {z} ) by A8, EQREL_1:def_4; y c= f " {(f . x)} by A2, A5, Th6; hence f " {(f . x)} = f " {z} by A7, A10, A9, XBOOLE_0:3; ::_thesis: verum end; consider h1 being Function such that A11: ( dom h1 = the carrier of (T_1-reflex T) & ( for y being set st y in the carrier of (T_1-reflex T) holds S1[y,h1 . y] ) ) from CLASSES1:sch_1(A4); defpred S2[ set , set ] means for z being Element of T1 st z in rng f & \$1 = f " {z} holds \$2 = z; A12: for y being set st y in fx holds ex w being set st S2[y,w] proof let y be set ; ::_thesis: ( y in fx implies ex w being set st S2[y,w] ) assume y in fx ; ::_thesis: ex w being set st S2[y,w] then consider w being Element of T1 such that A13: y = f " {w} and w in rng f ; take w ; ::_thesis: S2[y,w] let z be Element of T1; ::_thesis: ( z in rng f & y = f " {z} implies w = z ) assume that A14: z in rng f and A15: y = f " {z} ; ::_thesis: w = z now__::_thesis:_not_z_<>_w assume A16: z <> w ; ::_thesis: contradiction consider v being set such that A17: v in dom f and A18: z = f . v by A14, FUNCT_1:def_3; z in {z} by TARSKI:def_1; then v in f " {w} by A13, A15, A17, A18, FUNCT_1:def_7; then f . v in {w} by FUNCT_1:def_7; hence contradiction by A16, A18, TARSKI:def_1; ::_thesis: verum end; hence w = z ; ::_thesis: verum end; consider h2 being Function such that A19: ( dom h2 = fx & ( for y being set st y in fx holds S2[y,h2 . y] ) ) from CLASSES1:sch_1(A12); set h = h2 * h1; A20: dom (h2 * h1) = the carrier of (T_1-reflex T) proof thus dom (h2 * h1) c= the carrier of (T_1-reflex T) by A11, RELAT_1:25; :: according to XBOOLE_0:def_10 ::_thesis: the carrier of (T_1-reflex T) c= dom (h2 * h1) let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in the carrier of (T_1-reflex T) or z in dom (h2 * h1) ) assume A21: z in the carrier of (T_1-reflex T) ; ::_thesis: z in dom (h2 * h1) then consider w being Element of T1 such that A22: w in rng f and A23: z c= f " {w} by A2, Th7; h1 . z = f " {w} by A11, A21, A22, A23; then h1 . z in dom h2 by A19, A22; hence z in dom (h2 * h1) by A11, A21, FUNCT_1:11; ::_thesis: verum end; A24: dom ((h2 * h1) * (T_1-reflect T)) = the carrier of T proof thus dom ((h2 * h1) * (T_1-reflect T)) c= the carrier of T by A1, RELAT_1:25; :: according to XBOOLE_0:def_10 ::_thesis: the carrier of T c= dom ((h2 * h1) * (T_1-reflect T)) let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of T or y in dom ((h2 * h1) * (T_1-reflect T)) ) assume A25: y in the carrier of T ; ::_thesis: y in dom ((h2 * h1) * (T_1-reflect T)) then (T_1-reflect T) . y in rng (T_1-reflect T) by A1, FUNCT_1:def_3; hence y in dom ((h2 * h1) * (T_1-reflect T)) by A1, A20, A25, FUNCT_1:11; ::_thesis: verum end; A26: for x being set st x in dom f holds f . x = ((h2 * h1) * (T_1-reflect T)) . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = ((h2 * h1) * (T_1-reflect T)) . x ) assume A27: x in dom f ; ::_thesis: f . x = ((h2 * h1) * (T_1-reflect T)) . x then (T_1-reflect T) . x in rng (T_1-reflect T) by A1, FUNCT_1:def_3; then (T_1-reflect T) . x in the carrier of (T_1-reflex T) ; then (T_1-reflect T) . x in Intersection (Closed_Partitions T) by BORSUK_1:def_7; then consider y being Element of T such that A28: (T_1-reflect T) . x = EqClass (y,(Intersection (Closed_Partitions T))) by EQREL_1:42; reconsider x = x as Element of T by A27; reconsider fix = f . x as Element of T1 ; A29: x in EqClass (x,(Intersection (Closed_Partitions T))) by EQREL_1:def_6; T_1-reflect T = proj (Intersection (Closed_Partitions T)) by BORSUK_1:def_8; then x in (T_1-reflect T) . x by EQREL_1:def_9; then EqClass (x,(Intersection (Closed_Partitions T))) meets EqClass (y,(Intersection (Closed_Partitions T))) by A28, A29, XBOOLE_0:3; then A30: (T_1-reflect T) . x c= f " {fix} by A2, A28, Th6, EQREL_1:41; A31: fix in rng f by A27, FUNCT_1:def_3; then A32: f " {fix} in fx ; ((h2 * h1) * (T_1-reflect T)) . x = (h2 * h1) . ((T_1-reflect T) . x) by A24, FUNCT_1:12 .= h2 . (h1 . ((T_1-reflect T) . x)) by A11, FUNCT_1:13 .= h2 . (f " {fix}) by A11, A31, A30 .= f . x by A19, A31, A32 ; hence f . x = ((h2 * h1) * (T_1-reflect T)) . x ; ::_thesis: verum end; then A33: f = (h2 * h1) * (T_1-reflect T) by A3, A24, FUNCT_1:2; A34: rng h2 c= the carrier of T1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng h2 or y in the carrier of T1 ) assume y in rng h2 ; ::_thesis: y in the carrier of T1 then consider w being set such that A35: w in dom h2 and A36: y = h2 . w by FUNCT_1:def_3; consider x being Element of T1 such that A37: ( w = f " {x} & x in rng f ) by A19, A35; h2 . w = x by A19, A35, A37; hence y in the carrier of T1 by A36; ::_thesis: verum end; rng (h2 * h1) c= rng h2 proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng (h2 * h1) or z in rng h2 ) thus ( not z in rng (h2 * h1) or z in rng h2 ) by FUNCT_1:14; ::_thesis: verum end; then rng (h2 * h1) c= the carrier of T1 by A34, XBOOLE_1:1; then reconsider h = h2 * h1 as Function of the carrier of (T_1-reflex T), the carrier of T1 by A20, FUNCT_2:def_1, RELSET_1:4; reconsider h = h as Function of (T_1-reflex T),T1 ; h is continuous proof let y be Subset of T1; :: according to PRE_TOPC:def_6 ::_thesis: ( not y is closed or h " y is closed ) reconsider hy = h " y as Subset of (space (Intersection (Closed_Partitions T))) ; union hy c= the carrier of T proof let z1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not z1 in union hy or z1 in the carrier of T ) assume z1 in union hy ; ::_thesis: z1 in the carrier of T then consider z2 being set such that A38: z1 in z2 and A39: z2 in hy by TARSKI:def_4; z2 in the carrier of (space (Intersection (Closed_Partitions T))) by A39; then z2 in Intersection (Closed_Partitions T) by BORSUK_1:def_7; hence z1 in the carrier of T by A38; ::_thesis: verum end; then reconsider uhy = union hy as Subset of T ; assume y is closed ; ::_thesis: h " y is closed then (h * (T_1-reflect T)) " y is closed by A33, PRE_TOPC:def_6; then (T_1-reflect T) " (h " y) is closed by RELAT_1:146; then uhy is closed by Th1; hence h " y is closed by Th2; ::_thesis: verum end; then reconsider h = h as continuous Function of (T_1-reflex T),T1 ; take h ; ::_thesis: f = h * (T_1-reflect T) thus f = h * (T_1-reflect T) by A3, A24, A26, FUNCT_1:2; ::_thesis: verum end; definition let T, S be non empty TopSpace; let f be continuous Function of T,S; func T_1-reflex f -> continuous Function of (T_1-reflex T),(T_1-reflex S) means :: T_1TOPSP:def 4 (T_1-reflect S) * f = it * (T_1-reflect T); existence ex b1 being continuous Function of (T_1-reflex T),(T_1-reflex S) st (T_1-reflect S) * f = b1 * (T_1-reflect T) by Th8; uniqueness for b1, b2 being continuous Function of (T_1-reflex T),(T_1-reflex S) st (T_1-reflect S) * f = b1 * (T_1-reflect T) & (T_1-reflect S) * f = b2 * (T_1-reflect T) holds b1 = b2 proof let g1, g2 be continuous Function of (T_1-reflex T),(T_1-reflex S); ::_thesis: ( (T_1-reflect S) * f = g1 * (T_1-reflect T) & (T_1-reflect S) * f = g2 * (T_1-reflect T) implies g1 = g2 ) assume A1: ( (T_1-reflect S) * f = g1 * (T_1-reflect T) & (T_1-reflect S) * f = g2 * (T_1-reflect T) ) ; ::_thesis: g1 = g2 A2: now__::_thesis:_for_x_being_set_st_x_in_dom_g1_holds_ g2_._x_=_g1_._x let x be set ; ::_thesis: ( x in dom g1 implies g2 . x = g1 . x ) assume A3: x in dom g1 ; ::_thesis: g2 . x = g1 . x then A4: x in the carrier of (T_1-reflex T) ; A5: the carrier of (T_1-reflex T) = Intersection (Closed_Partitions T) by BORSUK_1:def_7; then consider y being Element of T such that A6: x = EqClass (y,(Intersection (Closed_Partitions T))) by A3, EQREL_1:42; reconsider y = y as Element of T ; set ty = (T_1-reflect T) . y; (T_1-reflect T) . y in Intersection (Closed_Partitions T) by A5; then A7: ( (T_1-reflect T) . y misses x or (T_1-reflect T) . y = x ) by A4, A5, EQREL_1:def_4; T_1-reflect T = proj (Intersection (Closed_Partitions T)) by BORSUK_1:def_8; then A8: ( dom (T_1-reflect T) = the carrier of T & y in (T_1-reflect T) . y ) by EQREL_1:def_9, FUNCT_2:def_1; A9: y in x by A6, EQREL_1:def_6; hence g2 . x = (g2 * (T_1-reflect T)) . y by A8, A7, FUNCT_1:13, XBOOLE_0:3 .= g1 . x by A1, A8, A9, A7, FUNCT_1:13, XBOOLE_0:3 ; ::_thesis: verum end; ( dom g1 = the carrier of (T_1-reflex T) & dom g2 = the carrier of (T_1-reflex T) ) by FUNCT_2:def_1; hence g1 = g2 by A2, FUNCT_1:2; ::_thesis: verum end; end; :: deftheorem defines T_1-reflex T_1TOPSP:def_4_:_ for T, S being non empty TopSpace for f being continuous Function of T,S for b4 being continuous Function of (T_1-reflex T),(T_1-reflex S) holds ( b4 = T_1-reflex f iff (T_1-reflect S) * f = b4 * (T_1-reflect T) );