:: TOPMETR semantic presentation

theorem Th1: :: TOPMETR:1
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is_a_cover_of b1 iff the carrier of b1 c= union b2 )
proof end;

theorem Th2: :: TOPMETR:2
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1
for b3 being Point of b2 holds
b3 is Point of b1
proof end;

theorem Th3: :: TOPMETR:3
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1 holds
( b1 is_T2 implies b2 is_T2 )
proof end;

theorem Th4: :: TOPMETR:4
for b1 being TopSpace
for b2, b3 being SubSpace of b1 holds
( the carrier of b2 c= the carrier of b3 implies b2 is SubSpace of b3 )
proof end;

theorem Th5: :: TOPMETR:5
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b1 | b2 is SubSpace of b1 | (b2 \/ b3) & b1 | b3 is SubSpace of b1 | (b2 \/ b3) )
proof end;

theorem Th6: :: TOPMETR:6
for b1 being non empty TopSpace
for b2 being Point of b1
for b3 being non empty Subset of b1 holds
( b2 in b3 implies for b4 being a_neighborhood of b2
for b5 being Point of (b1 | b3)
for b6 being Subset of (b1 | b3) holds
( b6 = b4 /\ b3 & b5 = b2 implies b6 is a_neighborhood of b5 ) )
proof end;

theorem Th7: :: TOPMETR:7
for b1, b2, b3 being TopSpace
for b4 being Function of b1,b3 holds
( b4 is continuous & b3 is SubSpace of b2 implies for b5 being Function of b1,b2 holds
( b5 = b4 implies b5 is continuous ) )
proof end;

theorem Th8: :: TOPMETR:8
for b1 being TopSpace
for b2 being non empty TopSpace
for b3 being Function of b1,b2
for b4 being SubSpace of b2 holds
( b3 is continuous implies for b5 being Function of b1,b4 holds
( b5 = b3 implies b5 is continuous ) )
proof end;

theorem Th9: :: TOPMETR:9
for b1, b2 being TopSpace
for b3 being Function of b1,b2
for b4 being Subset of b2 holds
( b3 is continuous implies for b5 being Function of b1,(b2 | b4) holds
( b5 = b3 implies b5 is continuous ) )
proof end;

theorem Th10: :: TOPMETR:10
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Subset of b1
for b4 being Function of b1,b2
for b5 being Function of (b1 | b3),b2 holds
( b4 is continuous & b5 = b4 | b3 implies b5 is continuous )
proof end;

Lemma5: for b1 being non empty MetrSpace
for b2 being Point of b1
for b3 being real number holds
( b3 > 0 implies b2 in Ball b2,b3 )
by TBSP_1:16;

Lemma6: for b1 being MetrSpace holds
MetrStruct(# the carrier of b1,the distance of b1 #) is MetrSpace
proof end;

definition
let c1 be MetrSpace;
mode SubSpace of c1 -> MetrSpace means :Def1: :: TOPMETR:def 1
( the carrier of a2 c= the carrier of a1 & ( for b1, b2 being Point of a2 holds the distance of a2 . b1,b2 = the distance of a1 . b1,b2 ) );
existence
ex b1 being MetrSpace st
( the carrier of b1 c= the carrier of c1 & ( for b2, b3 being Point of b1 holds the distance of b1 . b2,b3 = the distance of c1 . b2,b3 ) )
proof end;
end;

:: deftheorem Def1 defines SubSpace TOPMETR:def 1 :
for b1, b2 being MetrSpace holds
( b2 is SubSpace of b1 iff ( the carrier of b2 c= the carrier of b1 & ( for b3, b4 being Point of b2 holds the distance of b2 . b3,b4 = the distance of b1 . b3,b4 ) ) );

registration
let c1 be MetrSpace;
cluster strict SubSpace of a1;
existence
ex b1 being SubSpace of c1 st b1 is strict
proof end;
end;

registration
let c1 be non empty MetrSpace;
cluster non empty strict SubSpace of a1;
existence
ex b1 being SubSpace of c1 st
( b1 is strict & not b1 is empty )
proof end;
end;

theorem Th11: :: TOPMETR:11
canceled;

theorem Th12: :: TOPMETR:12
for b1 being non empty MetrSpace
for b2 being non empty SubSpace of b1
for b3 being Point of b2 holds
b3 is Point of b1
proof end;

theorem Th13: :: TOPMETR:13
for b1 being real number
for b2 being MetrSpace
for b3 being SubSpace of b2
for b4 being Point of b2
for b5 being Point of b3 holds
( b4 = b5 implies Ball b5,b1 = (Ball b4,b1) /\ the carrier of b3 )
proof end;

definition
let c1 be non empty MetrSpace;
let c2 be non empty Subset of c1;
func c1 | c2 -> strict SubSpace of a1 means :Def2: :: TOPMETR:def 2
the carrier of a3 = a2;
existence
ex b1 being strict SubSpace of c1 st the carrier of b1 = c2
proof end;
uniqueness
for b1, b2 being strict SubSpace of c1 holds
( the carrier of b1 = c2 & the carrier of b2 = c2 implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines | TOPMETR:def 2 :
for b1 being non empty MetrSpace
for b2 being non empty Subset of b1
for b3 being strict SubSpace of b1 holds
( b3 = b1 | b2 iff the carrier of b3 = b2 );

registration
let c1 be non empty MetrSpace;
let c2 be non empty Subset of c1;
cluster a1 | a2 -> non empty strict ;
coherence
not c1 | c2 is empty
proof end;
end;

definition
let c1, c2 be real number ;
assume E11: c1 <= c2 ;
func Closed-Interval-MSpace c1,c2 -> non empty strict SubSpace of RealSpace means :Def3: :: TOPMETR:def 3
for b1 being non empty Subset of RealSpace holds
( b1 = [.a1,a2.] implies a3 = RealSpace | b1 );
existence
ex b1 being non empty strict SubSpace of RealSpace st
for b2 being non empty Subset of RealSpace holds
( b2 = [.c1,c2.] implies b1 = RealSpace | b2 )
proof end;
uniqueness
for b1, b2 being non empty strict SubSpace of RealSpace holds
( ( for b3 being non empty Subset of RealSpace holds
( b3 = [.c1,c2.] implies b1 = RealSpace | b3 ) ) & ( for b3 being non empty Subset of RealSpace holds
( b3 = [.c1,c2.] implies b2 = RealSpace | b3 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines Closed-Interval-MSpace TOPMETR:def 3 :
for b1, b2 being real number holds
( b1 <= b2 implies for b3 being non empty strict SubSpace of RealSpace holds
( b3 = Closed-Interval-MSpace b1,b2 iff for b4 being non empty Subset of RealSpace holds
( b4 = [.b1,b2.] implies b3 = RealSpace | b4 ) ) );

theorem Th14: :: TOPMETR:14
for b1, b2 being real number holds
( b1 <= b2 implies the carrier of (Closed-Interval-MSpace b1,b2) = [.b1,b2.] )
proof end;

definition
let c1 be MetrStruct ;
let c2 be Subset-Family of c1;
attr a2 is being_ball-family means :Def4: :: TOPMETR:def 4
for b1 being set holds
not ( b1 in a2 & ( for b2 being Point of a1
for b3 being Real holds
not b1 = Ball b2,b3 ) );
pred c2 is_a_cover_of c1 means :Def5: :: TOPMETR:def 5
the carrier of a1 c= union a2;
end;

:: deftheorem Def4 defines being_ball-family TOPMETR:def 4 :
for b1 being MetrStruct
for b2 being Subset-Family of b1 holds
( b2 is being_ball-family iff for b3 being set holds
not ( b3 in b2 & ( for b4 being Point of b1
for b5 being Real holds
not b3 = Ball b4,b5 ) ) );

:: deftheorem Def5 defines is_a_cover_of TOPMETR:def 5 :
for b1 being MetrStruct
for b2 being Subset-Family of b1 holds
( b2 is_a_cover_of b1 iff the carrier of b1 c= union b2 );

notation
let c1 be MetrStruct ;
let c2 be Subset-Family of c1;
synonym c2 is_ball-family for being_ball-family c2;
end;

theorem Th15: :: TOPMETR:15
for b1, b2 being Point of RealSpace
for b3, b4 being real number holds
( b3 = b1 & b4 = b2 implies dist b1,b2 = abs (b3 - b4) )
proof end;

theorem Th16: :: TOPMETR:16
for b1 being MetrStruct holds
( the carrier of b1 = the carrier of (TopSpaceMetr b1) & the topology of (TopSpaceMetr b1) = Family_open_set b1 )
proof end;

theorem Th17: :: TOPMETR:17
canceled;

theorem Th18: :: TOPMETR:18
canceled;

theorem Th19: :: TOPMETR:19
for b1 being non empty MetrSpace
for b2 being non empty SubSpace of b1 holds
TopSpaceMetr b2 is SubSpace of TopSpaceMetr b1
proof end;

theorem Th20: :: TOPMETR:20
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3 being non empty Subset of (Euclid b1) holds
( b2 = b3 implies (TOP-REAL b1) | b2 = TopSpaceMetr ((Euclid b1) | b3) )
proof end;

theorem Th21: :: TOPMETR:21
for b1 being real number
for b2 being triangle MetrStruct
for b3 being Point of b2
for b4 being Subset of (TopSpaceMetr b2) holds
( b4 = Ball b3,b1 implies b4 is open )
proof end;

theorem Th22: :: TOPMETR:22
for b1 being non empty MetrSpace
for b2 being Subset of (TopSpaceMetr b1) holds
( b2 is open iff for b3 being Point of b1 holds
not ( b3 in b2 & ( for b4 being real number holds
not ( b4 > 0 & Ball b3,b4 c= b2 ) ) ) )
proof end;

definition
let c1 be MetrStruct ;
attr a1 is compact means :Def6: :: TOPMETR:def 6
TopSpaceMetr a1 is compact;
end;

:: deftheorem Def6 defines compact TOPMETR:def 6 :
for b1 being MetrStruct holds
( b1 is compact iff TopSpaceMetr b1 is compact );

theorem Th23: :: TOPMETR:23
for b1 being non empty MetrSpace holds
( b1 is compact iff for b2 being Subset-Family of b1 holds
not ( b2 is_ball-family & b2 is_a_cover_of b1 & ( for b3 being Subset-Family of b1 holds
not ( b3 c= b2 & b3 is_a_cover_of b1 & b3 is finite ) ) ) )
proof end;

definition
func R^1 -> strict TopSpace equals :: TOPMETR:def 7
TopSpaceMetr RealSpace ;
correctness
coherence
TopSpaceMetr RealSpace is strict TopSpace
;
;
end;

:: deftheorem Def7 defines R^1 TOPMETR:def 7 :
R^1 = TopSpaceMetr RealSpace ;

registration
cluster R^1 -> non empty strict ;
coherence
not R^1 is empty
;
end;

theorem Th24: :: TOPMETR:24
the carrier of R^1 = REAL
proof end;

registration
let c1 be set ;
let c2 be PartFunc of c1,the carrier of R^1 ;
let c3 be set ;
cluster a2 . a3 -> real ;
coherence
c2 . c3 is real
proof end;
end;

definition
let c1, c2 be real number ;
func Closed-Interval-TSpace c1,c2 -> non empty strict SubSpace of R^1 equals :: TOPMETR:def 8
TopSpaceMetr (Closed-Interval-MSpace a1,a2);
coherence
TopSpaceMetr (Closed-Interval-MSpace c1,c2) is non empty strict SubSpace of R^1
by Th19;
end;

:: deftheorem Def8 defines Closed-Interval-TSpace TOPMETR:def 8 :
for b1, b2 being real number holds Closed-Interval-TSpace b1,b2 = TopSpaceMetr (Closed-Interval-MSpace b1,b2);

theorem Th25: :: TOPMETR:25
for b1, b2 being real number holds
( b1 <= b2 implies the carrier of (Closed-Interval-TSpace b1,b2) = [.b1,b2.] )
proof end;

theorem Th26: :: TOPMETR:26
for b1, b2 being real number holds
( b1 <= b2 implies for b3 being Subset of R^1 holds
( b3 = [.b1,b2.] implies Closed-Interval-TSpace b1,b2 = R^1 | b3 ) )
proof end;

theorem Th27: :: TOPMETR:27
Closed-Interval-TSpace 0,1 = I[01]
proof end;

definition
redefine func I[01] as I[01] -> strict SubSpace of R^1 ;
coherence
I[01] is strict SubSpace of R^1
by Th27;
end;

Lemma25: for b1, b2, b3 being real number holds
( b3 >= 0 & b1 + b3 <= b2 implies b1 <= b2 )
proof end;

theorem Th28: :: TOPMETR:28
for b1 being Function of R^1 ,R^1 holds
( ex b2, b3 being Real st
for b4 being Real holds b1 . b4 = (b2 * b4) + b3 implies b1 is continuous )
proof end;

theorem Th29: :: TOPMETR:29
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b2 c= b3 implies b1 | b2 is SubSpace of b1 | b3 )
proof end;

theorem Th30: :: TOPMETR:30
for b1, b2, b3, b4 being real number
for b5 being Subset of (Closed-Interval-TSpace b3,b4) holds
( b3 <= b1 & b1 <= b2 & b2 <= b4 & b5 = [.b1,b2.] implies Closed-Interval-TSpace b1,b2 = (Closed-Interval-TSpace b3,b4) | b5 )
proof end;

theorem Th31: :: TOPMETR:31
for b1, b2 being real number
for b3 being Subset of I[01] holds
( 0 <= b1 & b1 <= b2 & b2 <= 1 & b3 = [.b1,b2.] implies Closed-Interval-TSpace b1,b2 = I[01] | b3 ) by Th30, Th27;