:: TOPS_2 semantic presentation

theorem Th1: :: TOPS_2:1
for b1 being 1-sorted
for b2 being Subset-Family of b1 holds b2 c= bool ([#] b1) ;

theorem Th2: :: TOPS_2:2
canceled;

theorem Th3: :: TOPS_2:3
for b1 being 1-sorted
for b2 being Subset-Family of b1
for b3 being set holds
( b3 c= b2 implies b3 is Subset-Family of b1 )
proof end;

theorem Th4: :: TOPS_2:4
canceled;

theorem Th5: :: TOPS_2:5
for b1 being non empty 1-sorted
for b2 being Subset-Family of b1 holds
not ( b2 is_a_cover_of b1 & not b2 <> {} )
proof end;

theorem Th6: :: TOPS_2:6
for b1 being 1-sorted
for b2, b3 being Subset-Family of b1 holds (union b2) \ (union b3) c= union (b2 \ b3)
proof end;

theorem Th7: :: TOPS_2:7
canceled;

theorem Th8: :: TOPS_2:8
canceled;

theorem Th9: :: TOPS_2:9
canceled;

theorem Th10: :: TOPS_2:10
for b1 being set
for b2 being Subset-Family of b1 holds
( not ( b2 <> {} & not COMPLEMENT b2 <> {} ) & not ( COMPLEMENT b2 <> {} & not b2 <> {} ) )
proof end;

theorem Th11: :: TOPS_2:11
for b1 being set
for b2 being Subset-Family of b1 holds
( b2 <> {} implies meet (COMPLEMENT b2) = (union b2) ` )
proof end;

theorem Th12: :: TOPS_2:12
for b1 being set
for b2 being Subset-Family of b1 holds
( b2 <> {} implies union (COMPLEMENT b2) = (meet b2) ` )
proof end;

theorem Th13: :: TOPS_2:13
for b1 being 1-sorted
for b2 being Subset-Family of b1 holds
( COMPLEMENT b2 is finite iff b2 is finite )
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset-Family of c1;
attr a2 is open means :Def1: :: TOPS_2:def 1
for b1 being Subset of a1 holds
( b1 in a2 implies b1 is open );
attr a2 is closed means :Def2: :: TOPS_2:def 2
for b1 being Subset of a1 holds
( b1 in a2 implies b1 is closed );
end;

:: deftheorem Def1 defines open TOPS_2:def 1 :
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is open iff for b3 being Subset of b1 holds
( b3 in b2 implies b3 is open ) );

:: deftheorem Def2 defines closed TOPS_2:def 2 :
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is closed iff for b3 being Subset of b1 holds
( b3 in b2 implies b3 is closed ) );

theorem Th14: :: TOPS_2:14
canceled;

theorem Th15: :: TOPS_2:15
canceled;

theorem Th16: :: TOPS_2:16
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is closed iff COMPLEMENT b2 is open )
proof end;

theorem Th17: :: TOPS_2:17
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is open iff COMPLEMENT b2 is closed )
proof end;

theorem Th18: :: TOPS_2:18
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 c= b3 & b3 is open implies b2 is open )
proof end;

theorem Th19: :: TOPS_2:19
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 c= b3 & b3 is closed implies b2 is closed )
proof end;

theorem Th20: :: TOPS_2:20
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 is open & b3 is open implies b2 \/ b3 is open )
proof end;

theorem Th21: :: TOPS_2:21
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 is open implies b2 /\ b3 is open )
proof end;

theorem Th22: :: TOPS_2:22
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 is open implies b2 \ b3 is open )
proof end;

theorem Th23: :: TOPS_2:23
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 is closed & b3 is closed implies b2 \/ b3 is closed )
proof end;

theorem Th24: :: TOPS_2:24
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 is closed implies b2 /\ b3 is closed )
proof end;

theorem Th25: :: TOPS_2:25
for b1 being TopStruct
for b2, b3 being Subset-Family of b1 holds
( b2 is closed implies b2 \ b3 is closed )
proof end;

theorem Th26: :: TOPS_2:26
for b1 being TopSpace
for b2 being Subset-Family of b1 holds
( b2 is open implies union b2 is open )
proof end;

theorem Th27: :: TOPS_2:27
for b1 being TopSpace
for b2 being Subset-Family of b1 holds
( b2 is open & b2 is finite implies meet b2 is open )
proof end;

theorem Th28: :: TOPS_2:28
for b1 being TopSpace
for b2 being Subset-Family of b1 holds
( b2 is closed & b2 is finite implies union b2 is closed )
proof end;

theorem Th29: :: TOPS_2:29
for b1 being TopSpace
for b2 being Subset-Family of b1 holds
( b2 is closed implies meet b2 is closed )
proof end;

theorem Th30: :: TOPS_2:30
canceled;

theorem Th31: :: TOPS_2:31
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Subset-Family of b2 holds
b3 is Subset-Family of b1
proof end;

theorem Th32: :: TOPS_2:32
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Subset of b2 holds
( b3 is open iff ex b4 being Subset of b1 st
( b4 is open & b4 /\ ([#] b2) = b3 ) )
proof end;

theorem Th33: :: TOPS_2:33
for b1 being TopStruct
for b2 being Subset of b1
for b3 being SubSpace of b1 holds
( b2 is open implies for b4 being Subset of b3 holds
( b4 = b2 implies b4 is open ) )
proof end;

theorem Th34: :: TOPS_2:34
for b1 being TopStruct
for b2 being Subset of b1
for b3 being SubSpace of b1 holds
( b2 is closed implies for b4 being Subset of b3 holds
( b4 = b2 implies b4 is closed ) )
proof end;

theorem Th35: :: TOPS_2:35
for b1 being TopStruct
for b2 being Subset-Family of b1
for b3 being SubSpace of b1 holds
( b2 is open implies for b4 being Subset-Family of b3 holds
( b4 = b2 implies b4 is open ) )
proof end;

theorem Th36: :: TOPS_2:36
for b1 being TopStruct
for b2 being Subset-Family of b1
for b3 being SubSpace of b1 holds
( b2 is closed implies for b4 being Subset-Family of b3 holds
( b4 = b2 implies b4 is closed ) )
proof end;

theorem Th37: :: TOPS_2:37
canceled;

theorem Th38: :: TOPS_2:38
for b1 being TopStruct
for b2, b3 being Subset of b1 holds
b2 /\ b3 is Subset of (b1 | b3)
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
let c3 be Subset-Family of c1;
func c3 | c2 -> Subset-Family of (a1 | a2) means :Def3: :: TOPS_2:def 3
for b1 being Subset of (a1 | a2) holds
( b1 in a4 iff ex b2 being Subset of a1 st
( b2 in a3 & b2 /\ a2 = b1 ) );
existence
ex b1 being Subset-Family of (c1 | c2) st
for b2 being Subset of (c1 | c2) holds
( b2 in b1 iff ex b3 being Subset of c1 st
( b3 in c3 & b3 /\ c2 = b2 ) )
proof end;
uniqueness
for b1, b2 being Subset-Family of (c1 | c2) holds
( ( for b3 being Subset of (c1 | c2) holds
( b3 in b1 iff ex b4 being Subset of c1 st
( b4 in c3 & b4 /\ c2 = b3 ) ) ) & ( for b3 being Subset of (c1 | c2) holds
( b3 in b2 iff ex b4 being Subset of c1 st
( b4 in c3 & b4 /\ c2 = b3 ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines | TOPS_2:def 3 :
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1
for b4 being Subset-Family of (b1 | b2) holds
( b4 = b3 | b2 iff for b5 being Subset of (b1 | b2) holds
( b5 in b4 iff ex b6 being Subset of b1 st
( b6 in b3 & b6 /\ b2 = b5 ) ) );

theorem Th39: :: TOPS_2:39
canceled;

theorem Th40: :: TOPS_2:40
for b1 being TopStruct
for b2 being Subset of b1
for b3, b4 being Subset-Family of b1 holds
( b3 c= b4 implies b3 | b2 c= b4 | b2 )
proof end;

theorem Th41: :: TOPS_2:41
for b1 being TopStruct
for b2, b3 being Subset of b1
for b4 being Subset-Family of b1 holds
( b2 in b4 implies b2 /\ b3 in b4 | b3 )
proof end;

theorem Th42: :: TOPS_2:42
for b1 being TopStruct
for b2, b3 being Subset of b1
for b4 being Subset-Family of b1 holds
( b2 c= union b4 implies b2 /\ b3 c= union (b4 | b3) )
proof end;

theorem Th43: :: TOPS_2:43
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1 holds
( b2 c= union b3 implies b2 = union (b3 | b2) )
proof end;

theorem Th44: :: TOPS_2:44
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1 holds union (b3 | b2) c= union b3
proof end;

theorem Th45: :: TOPS_2:45
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1 holds
( b2 c= union (b3 | b2) implies b2 c= union b3 )
proof end;

theorem Th46: :: TOPS_2:46
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1 holds
( b3 is finite implies b3 | b2 is finite )
proof end;

theorem Th47: :: TOPS_2:47
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1 holds
( b3 is open implies b3 | b2 is open )
proof end;

theorem Th48: :: TOPS_2:48
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1 holds
( b3 is closed implies b3 | b2 is closed )
proof end;

theorem Th49: :: TOPS_2:49
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Subset-Family of b2 holds
not ( b3 is open & ( for b4 being Subset-Family of b1 holds
not ( b4 is open & ( for b5 being Subset of b1 holds
( b5 = [#] b2 implies b3 = b4 | b5 ) ) ) ) )
proof end;

theorem Th50: :: TOPS_2:50
for b1 being TopStruct
for b2 being Subset of b1
for b3 being Subset-Family of b1 holds
ex b4 being Function st
( dom b4 = b3 & rng b4 = b3 | b2 & ( for b5 being set holds
( b5 in b3 implies for b6 being Subset of b1 holds
( b6 = b5 implies b4 . b5 = b6 /\ b2 ) ) ) )
proof end;

theorem Th51: :: TOPS_2:51
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function of b1,b2 holds
( dom b3 = [#] b1 & rng b3 c= [#] b2 ) by FUNCT_2:def 1;

theorem Th52: :: TOPS_2:52
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function of b1,b2 holds b3 " ([#] b2) = [#] b1
proof end;

theorem Th53: :: TOPS_2:53
canceled;

theorem Th54: :: TOPS_2:54
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function of b1,b2
for b4 being Subset-Family of b2 holds
(" b3) .: b4 is Subset-Family of b1
proof end;

theorem Th55: :: TOPS_2:55
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function of b1,b2 holds
( b3 is continuous iff for b4 being Subset of b2 holds
( b4 is open implies b3 " b4 is open ) )
proof end;

theorem Th56: :: TOPS_2:56
for b1, b2 being TopSpace
for b3 being Function of b1,b2 holds
( b3 is continuous iff for b4 being Subset of b2 holds Cl (b3 " b4) c= b3 " (Cl b4) )
proof end;

theorem Th57: :: TOPS_2:57
for b1 being TopSpace
for b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is continuous iff for b4 being Subset of b1 holds b3 .: (Cl b4) c= Cl (b3 .: b4) )
proof end;

theorem Th58: :: TOPS_2:58
for b1, b2 being TopStruct
for b3 being non empty TopStruct
for b4 being Function of b1,b3
for b5 being Function of b3,b2 holds
( b4 is continuous & b5 is continuous implies b5 * b4 is continuous )
proof end;

theorem Th59: :: TOPS_2:59
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function of b1,b2
for b4 being Subset-Family of b2 holds
( b3 is continuous & b4 is open implies for b5 being Subset-Family of b1 holds
( b5 = (" b3) .: b4 implies b5 is open ) )
proof end;

theorem Th60: :: TOPS_2:60
for b1, b2 being TopStruct
for b3 being Function of b1,b2
for b4 being Subset-Family of b2 holds
( b3 is continuous & b4 is closed implies for b5 being Subset-Family of b1 holds
( b5 = (" b3) .: b4 implies b5 is closed ) )
proof end;

definition
let c1, c2 be 1-sorted ;
let c3 be Function of c1,c2;
assume that
E23: rng c3 = [#] c2 and
E24: c3 is one-to-one ;
func c3 /" -> Function of a2,a1 equals :Def4: :: TOPS_2:def 4
a3 " ;
coherence
c3 " is Function of c2,c1
by E23, E24, FUNCT_2:31;
end;

:: deftheorem Def4 defines /" TOPS_2:def 4 :
for b1, b2 being 1-sorted
for b3 being Function of b1,b2 holds
( rng b3 = [#] b2 & b3 is one-to-one implies b3 /" = b3 " );

notation
let c1, c2 be 1-sorted ;
let c3 be Function of c1,c2;
synonym c3 " for c3 /" ;
end;

theorem Th61: :: TOPS_2:61
canceled;

theorem Th62: :: TOPS_2:62
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function of b1,b2 holds
( rng b3 = [#] b2 & b3 is one-to-one implies ( dom (b3 " ) = [#] b2 & rng (b3 " ) = [#] b1 ) )
proof end;

theorem Th63: :: TOPS_2:63
for b1, b2 being 1-sorted
for b3 being Function of b1,b2 holds
( rng b3 = [#] b2 & b3 is one-to-one implies b3 " is one-to-one )
proof end;

theorem Th64: :: TOPS_2:64
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function of b1,b2 holds
( rng b3 = [#] b2 & b3 is one-to-one implies (b3 " ) " = b3 )
proof end;

theorem Th65: :: TOPS_2:65
for b1, b2 being 1-sorted
for b3 being Function of b1,b2 holds
( rng b3 = [#] b2 & b3 is one-to-one implies ( (b3 " ) * b3 = id (dom b3) & b3 * (b3 " ) = id (rng b3) ) )
proof end;

theorem Th66: :: TOPS_2:66
for b1 being 1-sorted
for b2, b3 being non empty 1-sorted
for b4 being Function of b1,b2
for b5 being Function of b2,b3 holds
( dom b4 = [#] b1 & rng b4 = [#] b2 & b4 is one-to-one & dom b5 = [#] b2 & rng b5 = [#] b3 & b5 is one-to-one implies (b5 * b4) " = (b4 " ) * (b5 " ) )
proof end;

theorem Th67: :: TOPS_2:67
for b1, b2 being 1-sorted
for b3 being Function of b1,b2
for b4 being Subset of b1 holds
( rng b3 = [#] b2 & b3 is one-to-one implies b3 .: b4 = (b3 " ) " b4 )
proof end;

theorem Th68: :: TOPS_2:68
for b1, b2 being 1-sorted
for b3 being Function of b1,b2
for b4 being Subset of b2 holds
( rng b3 = [#] b2 & b3 is one-to-one implies b3 " b4 = (b3 " ) .: b4 )
proof end;

definition
let c1, c2 be TopStruct ;
let c3 be Function of c1,c2;
attr a3 is being_homeomorphism means :Def5: :: TOPS_2:def 5
( dom a3 = [#] a1 & rng a3 = [#] a2 & a3 is one-to-one & a3 is continuous & a3 " is continuous );
end;

:: deftheorem Def5 defines being_homeomorphism TOPS_2:def 5 :
for b1, b2 being TopStruct
for b3 being Function of b1,b2 holds
( b3 is being_homeomorphism iff ( dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & b3 is continuous & b3 " is continuous ) );

notation
let c1, c2 be TopStruct ;
let c3 be Function of c1,c2;
synonym c3 is_homeomorphism for being_homeomorphism c3;
end;

theorem Th69: :: TOPS_2:69
canceled;

theorem Th70: :: TOPS_2:70
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism implies b3 " is_homeomorphism )
proof end;

theorem Th71: :: TOPS_2:71
for b1, b2, b3 being non empty TopStruct
for b4 being Function of b1,b2
for b5 being Function of b2,b3 holds
( b4 is_homeomorphism & b5 is_homeomorphism implies b5 * b4 is_homeomorphism )
proof end;

theorem Th72: :: TOPS_2:72
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism iff ( dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & ( for b4 being Subset of b1 holds
( b4 is closed iff b3 .: b4 is closed ) ) ) )
proof end;

theorem Th73: :: TOPS_2:73
for b1 being non empty TopSpace
for b2 being TopSpace
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism iff ( dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & ( for b4 being Subset of b2 holds b3 " (Cl b4) = Cl (b3 " b4) ) ) )
proof end;

theorem Th74: :: TOPS_2:74
for b1 being TopSpace
for b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( b3 is_homeomorphism iff ( dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & ( for b4 being Subset of b1 holds b3 .: (Cl b4) = Cl (b3 .: b4) ) ) )
proof end;

theorem Th75: :: TOPS_2:75
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2
for b4 being Subset of b1 holds
( b3 is continuous & b4 is connected implies b3 .: b4 is connected )
proof end;

theorem Th76: :: TOPS_2:76
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2
for b4 being Subset of b2 holds
( b3 is_homeomorphism & b4 is connected implies b3 " b4 is connected )
proof end;