:: WAYBEL18 semantic presentation

theorem Th1: :: WAYBEL18:1
canceled;

theorem Th2: :: WAYBEL18:2
for b1 being set
for b2, b3 being Subset-Family of b1 holds
( ( b3 = b2 \ {{} } or b2 = b3 \/ {{} } ) implies UniCl b2 = UniCl b3 )
proof end;

theorem Th3: :: WAYBEL18:3
for b1 being TopSpace
for b2 being Subset-Family of b1 holds
( b2 is Basis of b1 iff b2 \ {{} } is Basis of b1 )
proof end;

definition
let c1 be Relation;
attr a1 is TopSpace-yielding means :Def1: :: WAYBEL18:def 1
for b1 being set holds
( b1 in rng a1 implies b1 is TopStruct );
end;

:: deftheorem Def1 defines TopSpace-yielding WAYBEL18:def 1 :
for b1 being Relation holds
( b1 is TopSpace-yielding iff for b2 being set holds
( b2 in rng b1 implies b2 is TopStruct ) );

registration
cluster TopSpace-yielding -> 1-sorted-yielding set ;
coherence
for b1 being Function holds
( b1 is TopSpace-yielding implies b1 is 1-sorted-yielding )
proof end;
end;

registration
let c1 be set ;
cluster 1-sorted-yielding TopSpace-yielding ManySortedSet of a1;
existence
ex b1 being ManySortedSet of c1 st b1 is TopSpace-yielding
proof end;
end;

registration
let c1 be set ;
cluster 1-sorted-yielding non-Empty TopSpace-yielding ManySortedSet of a1;
existence
ex b1 being ManySortedSet of c1 st
( b1 is TopSpace-yielding & b1 is non-Empty )
proof end;
end;

definition
let c1 be non empty set ;
let c2 be TopSpace-yielding ManySortedSet of c1;
let c3 be Element of c1;
redefine func . as c2 . c3 -> TopStruct ;
coherence
c2 . c3 is TopStruct
proof end;
end;

definition
let c1 be set ;
let c2 be TopSpace-yielding ManySortedSet of c1;
func product_prebasis c2 -> Subset-Family of (product (Carrier a2)) means :Def2: :: WAYBEL18:def 2
for b1 being Subset of (product (Carrier a2)) holds
( b1 in a3 iff ex b2 being set ex b3 being TopStruct ex b4 being Subset of b3 st
( b2 in a1 & b4 is open & b3 = a2 . b2 & b1 = product ((Carrier a2) +* b2,b4) ) );
existence
ex b1 being Subset-Family of (product (Carrier c2)) st
for b2 being Subset of (product (Carrier c2)) holds
( b2 in b1 iff ex b3 being set ex b4 being TopStruct ex b5 being Subset of b4 st
( b3 in c1 & b5 is open & b4 = c2 . b3 & b2 = product ((Carrier c2) +* b3,b5) ) )
proof end;
uniqueness
for b1, b2 being Subset-Family of (product (Carrier c2)) holds
( ( for b3 being Subset of (product (Carrier c2)) holds
( b3 in b1 iff ex b4 being set ex b5 being TopStruct ex b6 being Subset of b5 st
( b4 in c1 & b6 is open & b5 = c2 . b4 & b3 = product ((Carrier c2) +* b4,b6) ) ) ) & ( for b3 being Subset of (product (Carrier c2)) holds
( b3 in b2 iff ex b4 being set ex b5 being TopStruct ex b6 being Subset of b5 st
( b4 in c1 & b6 is open & b5 = c2 . b4 & b3 = product ((Carrier c2) +* b4,b6) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines product_prebasis WAYBEL18:def 2 :
for b1 being set
for b2 being TopSpace-yielding ManySortedSet of b1
for b3 being Subset-Family of (product (Carrier b2)) holds
( b3 = product_prebasis b2 iff for b4 being Subset of (product (Carrier b2)) holds
( b4 in b3 iff ex b5 being set ex b6 being TopStruct ex b7 being Subset of b6 st
( b5 in b1 & b7 is open & b6 = b2 . b5 & b4 = product ((Carrier b2) +* b5,b7) ) ) );

theorem Th4: :: WAYBEL18:4
for b1 being set
for b2 being Subset-Family of b1 holds TopStruct(# b1,(UniCl (FinMeetCl b2)) #) is TopSpace-like
proof end;

definition
let c1 be set ;
let c2 be non-Empty TopSpace-yielding ManySortedSet of c1;
func product c2 -> strict TopSpace means :Def3: :: WAYBEL18:def 3
( the carrier of a3 = product (Carrier a2) & product_prebasis a2 is prebasis of a3 );
existence
ex b1 being strict TopSpace st
( the carrier of b1 = product (Carrier c2) & product_prebasis c2 is prebasis of b1 )
proof end;
uniqueness
for b1, b2 being strict TopSpace holds
( the carrier of b1 = product (Carrier c2) & product_prebasis c2 is prebasis of b1 & the carrier of b2 = product (Carrier c2) & product_prebasis c2 is prebasis of b2 implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines product WAYBEL18:def 3 :
for b1 being set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being strict TopSpace holds
( b3 = product b2 iff ( the carrier of b3 = product (Carrier b2) & product_prebasis b2 is prebasis of b3 ) );

registration
let c1 be set ;
let c2 be non-Empty TopSpace-yielding ManySortedSet of c1;
cluster product a2 -> non empty strict ;
coherence
not product c2 is empty
proof end;
end;

definition
let c1 be non empty set ;
let c2 be non-Empty TopSpace-yielding ManySortedSet of c1;
let c3 be Element of c1;
redefine func . as c2 . c3 -> non empty TopStruct ;
coherence
c2 . c3 is non empty TopStruct
proof end;
end;

registration
let c1 be set ;
let c2 be non-Empty TopSpace-yielding ManySortedSet of c1;
cluster product a2 -> non empty strict constituted-Functions ;
coherence
product c2 is constituted-Functions
proof end;
end;

definition
let c1 be non empty set ;
let c2 be non-Empty TopSpace-yielding ManySortedSet of c1;
let c3 be Element of (product c2);
let c4 be Element of c1;
redefine func . as c3 . c4 -> Element of (a2 . a4);
coherence
c3 . c4 is Element of (c2 . c4)
proof end;
end;

definition
let c1 be non empty set ;
let c2 be non-Empty TopSpace-yielding ManySortedSet of c1;
let c3 be Element of c1;
func proj c2,c3 -> Function of (product a2),(a2 . a3) equals :: WAYBEL18:def 4
proj (Carrier a2),a3;
coherence
proj (Carrier c2),c3 is Function of (product c2),(c2 . c3)
proof end;
end;

:: deftheorem Def4 defines proj WAYBEL18:def 4 :
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1 holds proj b2,b3 = proj (Carrier b2),b3;

theorem Th5: :: WAYBEL18:5
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Subset of (b2 . b3) holds (proj b2,b3) " b4 = product ((Carrier b2) +* b3,b4)
proof end;

theorem Th6: :: WAYBEL18:6
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1 holds proj b2,b3 is continuous
proof end;

theorem Th7: :: WAYBEL18:7
for b1 being non empty TopSpace
for b2 being non empty set
for b3 being non-Empty TopSpace-yielding ManySortedSet of b2
for b4 being Function of b1,(product b3) holds
( b4 is continuous iff for b5 being Element of b2 holds (proj b3,b5) * b4 is continuous )
proof end;

definition
let c1 be TopStruct ;
attr a1 is injective means :Def5: :: WAYBEL18:def 5
for b1 being non empty TopSpace
for b2 being Function of b1,a1 holds
( b2 is continuous implies for b3 being non empty TopSpace holds
not ( b1 is SubSpace of b3 & ( for b4 being Function of b3,a1 holds
not ( b4 is continuous & b4 | the carrier of b1 = b2 ) ) ) );
end;

:: deftheorem Def5 defines injective WAYBEL18:def 5 :
for b1 being TopStruct holds
( b1 is injective iff for b2 being non empty TopSpace
for b3 being Function of b2,b1 holds
( b3 is continuous implies for b4 being non empty TopSpace holds
not ( b2 is SubSpace of b4 & ( for b5 being Function of b4,b1 holds
not ( b5 is continuous & b5 | the carrier of b2 = b3 ) ) ) ) );

theorem Th8: :: WAYBEL18:8
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1 holds
( ( for b3 being Element of b1 holds b2 . b3 is injective ) implies product b2 is injective )
proof end;

theorem Th9: :: WAYBEL18:9
for b1 being non empty TopSpace holds
( b1 is injective implies for b2 being non empty SubSpace of b1 holds
( b2 is_a_retract_of b1 implies b2 is injective ) )
proof end;

definition
let c1 be 1-sorted ;
let c2 be TopStruct ;
let c3 be Function of c1,c2;
func Image c3 -> SubSpace of a2 equals :: WAYBEL18:def 6
a2 | (rng a3);
coherence
c2 | (rng c3) is SubSpace of c2
;
end;

:: deftheorem Def6 defines Image WAYBEL18:def 6 :
for b1 being 1-sorted
for b2 being TopStruct
for b3 being Function of b1,b2 holds Image b3 = b2 | (rng b3);

registration
let c1 be non empty 1-sorted ;
let c2 be non empty TopStruct ;
let c3 be Function of c1,c2;
cluster Image a3 -> non empty ;
coherence
not Image c3 is empty
proof end;
end;

theorem Th10: :: WAYBEL18:10
for b1 being 1-sorted
for b2 being TopStruct
for b3 being Function of b1,b2 holds the carrier of (Image b3) = rng b3
proof end;

definition
let c1 be 1-sorted ;
let c2 be non empty TopStruct ;
let c3 be Function of c1,c2;
func corestr c3 -> Function of a1,(Image a3) equals :: WAYBEL18:def 7
a3;
coherence
c3 is Function of c1,(Image c3)
proof end;
end;

:: deftheorem Def7 defines corestr WAYBEL18:def 7 :
for b1 being 1-sorted
for b2 being non empty TopStruct
for b3 being Function of b1,b2 holds corestr b3 = b3;

theorem Th11: :: WAYBEL18:11
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is continuous implies corestr b3 is continuous )
proof end;

registration
let c1 be 1-sorted ;
let c2 be non empty TopStruct ;
let c3 be Function of c1,c2;
cluster corestr a3 -> onto ;
coherence
corestr c3 is onto
proof end;
end;

definition
let c1, c2 be TopStruct ;
pred c1 is_Retract_of c2 means :: WAYBEL18:def 8
ex b1 being Function of a2,a2 st
( b1 is continuous & b1 * b1 = b1 & Image b1,a1 are_homeomorphic );
end;

:: deftheorem Def8 defines is_Retract_of WAYBEL18:def 8 :
for b1, b2 being TopStruct holds
( b1 is_Retract_of b2 iff ex b3 being Function of b2,b2 st
( b3 is continuous & b3 * b3 = b3 & Image b3,b1 are_homeomorphic ) );

theorem Th12: :: WAYBEL18:12
for b1, b2 being non empty TopSpace holds
( b1 is injective implies for b3 being Function of b1,b2 holds
( corestr b3 is_homeomorphism implies b1 is_Retract_of b2 ) )
proof end;

definition
func Sierpinski_Space -> strict TopStruct means :Def9: :: WAYBEL18:def 9
( the carrier of a1 = {0,1} & the topology of a1 = {{} ,{1},{0,1}} );
existence
ex b1 being strict TopStruct st
( the carrier of b1 = {0,1} & the topology of b1 = {{} ,{1},{0,1}} )
proof end;
uniqueness
for b1, b2 being strict TopStruct holds
( the carrier of b1 = {0,1} & the topology of b1 = {{} ,{1},{0,1}} & the carrier of b2 = {0,1} & the topology of b2 = {{} ,{1},{0,1}} implies b1 = b2 )
;
end;

:: deftheorem Def9 defines Sierpinski_Space WAYBEL18:def 9 :
for b1 being strict TopStruct holds
( b1 = Sierpinski_Space iff ( the carrier of b1 = {0,1} & the topology of b1 = {{} ,{1},{0,1}} ) );

registration
cluster Sierpinski_Space -> non empty strict TopSpace-like ;
coherence
( not Sierpinski_Space is empty & Sierpinski_Space is TopSpace-like )
proof end;
end;

Lemma16: Sierpinski_Space is discerning
proof end;

registration
cluster Sierpinski_Space -> non empty strict TopSpace-like discerning ;
coherence
Sierpinski_Space is discerning
by Lemma16;
end;

registration
cluster Sierpinski_Space -> non empty strict TopSpace-like discerning injective ;
coherence
Sierpinski_Space is injective
proof end;
end;

registration
let c1 be set ;
let c2 be non empty 1-sorted ;
cluster a1 --> a2 -> non-Empty ;
coherence
c1 --> c2 is non-Empty
proof end;
end;

registration
let c1 be set ;
let c2 be TopStruct ;
cluster a1 --> a2 -> 1-sorted-yielding TopSpace-yielding ;
coherence
c1 --> c2 is TopSpace-yielding
proof end;
end;

registration
let c1 be set ;
let c2 be reflexive RelStr ;
cluster a1 --> a2 -> reflexive-yielding ;
coherence
c1 --> c2 is reflexive-yielding
proof end;
end;

registration
let c1 be non empty set ;
let c2 be non empty antisymmetric RelStr ;
cluster product (a1 --> a2) -> antisymmetric ;
coherence
product (c1 --> c2) is antisymmetric
proof end;
end;

registration
let c1 be non empty set ;
let c2 be non empty transitive RelStr ;
cluster product (a1 --> a2) -> transitive ;
coherence
product (c1 --> c2) is transitive
proof end;
end;

theorem Th13: :: WAYBEL18:13
for b1 being Scott TopAugmentation of BoolePoset 1 holds the topology of b1 = the topology of Sierpinski_Space
proof end;

theorem Th14: :: WAYBEL18:14
for b1 being non empty set holds
{ (product ((Carrier (b1 --> Sierpinski_Space )) +* b2,{1})) where B is Element of b1 : verum } is prebasis of product (b1 --> Sierpinski_Space )
proof end;

registration
let c1 be non empty set ;
let c2 be complete LATTICE;
cluster product (a1 --> a2) -> transitive antisymmetric with_suprema complete ;
coherence
( product (c1 --> c2) is with_suprema & product (c1 --> c2) is complete )
proof end;
end;

registration
let c1 be non empty set ;
let c2 be lower-bounded algebraic LATTICE;
cluster product (a1 --> a2) -> transitive antisymmetric with_suprema complete algebraic ;
coherence
product (c1 --> c2) is algebraic
proof end;
end;

theorem Th15: :: WAYBEL18:15
for b1 being non empty set holds
ex b2 being Function of (BoolePoset b1),(product (b1 --> (BoolePoset 1))) st
( b2 is isomorphic & ( for b3 being Subset of b1 holds b2 . b3 = chi b3,b1 ) )
proof end;

theorem Th16: :: WAYBEL18:16
for b1 being non empty set
for b2 being Scott TopAugmentation of product (b1 --> (BoolePoset 1)) holds the topology of b2 = the topology of (product (b1 --> Sierpinski_Space ))
proof end;

theorem Th17: :: WAYBEL18:17
for b1, b2 being non empty TopSpace holds
( the carrier of b1 = the carrier of b2 & the topology of b1 = the topology of b2 & b1 is injective implies b2 is injective )
proof end;

theorem Th18: :: WAYBEL18:18
for b1 being non empty set
for b2 being Scott TopAugmentation of product (b1 --> (BoolePoset 1)) holds b2 is injective
proof end;

theorem Th19: :: WAYBEL18:19
for b1 being T_0-TopSpace holds
ex b2 being non empty set ex b3 being Function of b1,(product (b2 --> Sierpinski_Space )) st corestr b3 is_homeomorphism
proof end;

theorem Th20: :: WAYBEL18:20
for b1 being T_0-TopSpace holds
not ( b1 is injective & ( for b2 being non empty set holds
not b1 is_Retract_of product (b2 --> Sierpinski_Space ) ) )
proof end;