:: CONNSP_1 semantic presentation

definition
let c1 be TopStruct ;
let c2, c3 be Subset of c1;
pred c2,c3 are_separated means :Def1: :: CONNSP_1:def 1
( Cl a2 misses a3 & a2 misses Cl a3 );
symmetry
for b1, b2 being Subset of c1 holds
( Cl b1 misses b2 & b1 misses Cl b2 implies ( Cl b2 misses b1 & b2 misses Cl b1 ) )
;
end;

:: deftheorem Def1 defines are_separated CONNSP_1:def 1 :
for b1 being TopStruct
for b2, b3 being Subset of b1 holds
( b2,b3 are_separated iff ( Cl b2 misses b3 & b2 misses Cl b3 ) );

theorem Th1: :: CONNSP_1:1
canceled;

theorem Th2: :: CONNSP_1:2
for b1 being TopStruct
for b2, b3 being Subset of b1 holds
( b2,b3 are_separated implies b2 misses b3 )
proof end;

theorem Th3: :: CONNSP_1:3
for b1 being TopStruct
for b2, b3 being Subset of b1 holds
( [#] b1 = b2 \/ b3 & b2 is closed & b3 is closed & b2 misses b3 implies b2,b3 are_separated )
proof end;

theorem Th4: :: CONNSP_1:4
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( [#] b1 = b2 \/ b3 & b2 is open & b3 is open & b2 misses b3 implies b2,b3 are_separated )
proof end;

theorem Th5: :: CONNSP_1:5
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( [#] b1 = b2 \/ b3 & b2,b3 are_separated implies ( b2 is open & b2 is closed & b3 is open & b3 is closed ) )
proof end;

theorem Th6: :: CONNSP_1:6
for b1 being TopSpace
for b2 being SubSpace of b1
for b3, b4 being Subset of b1
for b5, b6 being Subset of b2 holds
( b5 = b3 & b6 = b4 & b5,b6 are_separated implies b3,b4 are_separated )
proof end;

theorem Th7: :: CONNSP_1:7
for b1 being TopSpace
for b2 being SubSpace of b1
for b3, b4 being Subset of b1
for b5, b6 being Subset of b2 holds
( b3 = b5 & b4 = b6 & b3 \/ b4 c= [#] b2 & b3,b4 are_separated implies b5,b6 are_separated )
proof end;

theorem Th8: :: CONNSP_1:8
for b1 being TopStruct
for b2, b3, b4, b5 being Subset of b1 holds
( b2,b3 are_separated & b4 c= b2 & b5 c= b3 implies b4,b5 are_separated )
proof end;

theorem Th9: :: CONNSP_1:9
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2,b3 are_separated & b2,b4 are_separated implies b2,b3 \/ b4 are_separated )
proof end;

theorem Th10: :: CONNSP_1:10
for b1 being TopStruct
for b2, b3 being Subset of b1 holds
( ( ( b2 is closed & b3 is closed ) or ( b2 is open & b3 is open ) ) implies b2 \ b3,b3 \ b2 are_separated )
proof end;

definition
let c1 be TopStruct ;
attr a1 is connected means :Def2: :: CONNSP_1:def 2
for b1, b2 being Subset of a1 holds
not ( [#] a1 = b1 \/ b2 & b1,b2 are_separated & not b1 = {} a1 & not b2 = {} a1 );
end;

:: deftheorem Def2 defines connected CONNSP_1:def 2 :
for b1 being TopStruct holds
( b1 is connected iff for b2, b3 being Subset of b1 holds
not ( [#] b1 = b2 \/ b3 & b2,b3 are_separated & not b2 = {} b1 & not b3 = {} b1 ) );

theorem Th11: :: CONNSP_1:11
for b1 being TopSpace holds
( b1 is connected iff for b2, b3 being Subset of b1 holds
not ( [#] b1 = b2 \/ b3 & b2 <> {} b1 & b3 <> {} b1 & b2 is closed & b3 is closed & not b2 meets b3 ) )
proof end;

theorem Th12: :: CONNSP_1:12
for b1 being TopSpace holds
( b1 is connected iff for b2, b3 being Subset of b1 holds
not ( [#] b1 = b2 \/ b3 & b2 <> {} b1 & b3 <> {} b1 & b2 is open & b3 is open & not b2 meets b3 ) )
proof end;

theorem Th13: :: CONNSP_1:13
for b1 being TopSpace holds
( b1 is connected iff for b2 being Subset of b1 holds
not ( b2 <> {} b1 & b2 <> [#] b1 & not Cl b2 meets Cl (([#] b1) \ b2) ) )
proof end;

theorem Th14: :: CONNSP_1:14
for b1 being TopSpace holds
( b1 is connected iff for b2 being Subset of b1 holds
not ( b2 is open & b2 is closed & not b2 = {} b1 & not b2 = [#] b1 ) )
proof end;

theorem Th15: :: CONNSP_1:15
for b1, b2 being TopSpace
for b3 being Function of b1,b2 holds
( b3 is continuous & b3 .: ([#] b1) = [#] b2 & b1 is connected implies b2 is connected )
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
attr a2 is connected means :Def3: :: CONNSP_1:def 3
a1 | a2 is connected;
end;

:: deftheorem Def3 defines connected CONNSP_1:def 3 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is connected iff b1 | b2 is connected );

theorem Th16: :: CONNSP_1:16
for b1 being TopSpace
for b2 being Subset of b1 holds
( b2 is connected iff for b3, b4 being Subset of b1 holds
not ( b2 = b3 \/ b4 & b3,b4 are_separated & not b3 = {} b1 & not b4 = {} b1 ) )
proof end;

theorem Th17: :: CONNSP_1:17
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
not ( b2 is connected & b2 c= b3 \/ b4 & b3,b4 are_separated & not b2 c= b3 & not b2 c= b4 )
proof end;

theorem Th18: :: CONNSP_1:18
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is connected & b3 is connected & not b2,b3 are_separated implies b2 \/ b3 is connected )
proof end;

theorem Th19: :: CONNSP_1:19
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is connected & b2 c= b3 & b3 c= Cl b2 implies b3 is connected )
proof end;

theorem Th20: :: CONNSP_1:20
for b1 being TopSpace
for b2 being Subset of b1 holds
( b2 is connected implies Cl b2 is connected )
proof end;

theorem Th21: :: CONNSP_1:21
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( b1 is connected & b2 is connected & ([#] b1) \ b2 = b3 \/ b4 & b3,b4 are_separated implies ( b2 \/ b3 is connected & b2 \/ b4 is connected ) )
proof end;

theorem Th22: :: CONNSP_1:22
for b1 being TopSpace
for b2, b3, b4 being Subset of b1 holds
( ([#] b1) \ b2 = b3 \/ b4 & b3,b4 are_separated & b2 is closed implies ( b2 \/ b3 is closed & b2 \/ b4 is closed ) )
proof end;

theorem Th23: :: CONNSP_1:23
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
not ( b2 is connected & b2 meets b3 & b2 \ b3 <> {} b1 & not b2 meets Fr b3 )
proof end;

theorem Th24: :: CONNSP_1:24
for b1 being TopSpace
for b2 being SubSpace of b1
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 implies ( b3 is connected iff b4 is connected ) )
proof end;

theorem Th25: :: CONNSP_1:25
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is closed & b3 is closed & b2 \/ b3 is connected & b2 /\ b3 is connected implies ( b2 is connected & b3 is connected ) )
proof end;

theorem Th26: :: CONNSP_1:26
for b1 being TopSpace
for b2 being Subset-Family of b1 holds
( ( for b3 being Subset of b1 holds
( b3 in b2 implies b3 is connected ) ) & ex b3 being Subset of b1 st
( b3 <> {} b1 & b3 in b2 & ( for b4 being Subset of b1 holds
not ( b4 in b2 & b4 <> b3 & b3,b4 are_separated ) ) ) implies union b2 is connected )
proof end;

theorem Th27: :: CONNSP_1:27
for b1 being TopSpace
for b2 being Subset-Family of b1 holds
( ( for b3 being Subset of b1 holds
( b3 in b2 implies b3 is connected ) ) & meet b2 <> {} b1 implies union b2 is connected )
proof end;

theorem Th28: :: CONNSP_1:28
for b1 being TopSpace holds
( [#] b1 is connected iff b1 is connected )
proof end;

theorem Th29: :: CONNSP_1:29
for b1 being non empty TopSpace
for b2 being Point of b1 holds {b2} is connected
proof end;

definition
let c1 be TopStruct ;
let c2, c3 be Point of c1;
pred c2,c3 are_joined means :Def4: :: CONNSP_1:def 4
ex b1 being Subset of a1 st
( b1 is connected & a2 in b1 & a3 in b1 );
end;

:: deftheorem Def4 defines are_joined CONNSP_1:def 4 :
for b1 being TopStruct
for b2, b3 being Point of b1 holds
( b2,b3 are_joined iff ex b4 being Subset of b1 st
( b4 is connected & b2 in b4 & b3 in b4 ) );

theorem Th30: :: CONNSP_1:30
for b1 being non empty TopSpace holds
( ex b2 being Point of b1 st
for b3 being Point of b1 holds b2,b3 are_joined implies b1 is connected )
proof end;

theorem Th31: :: CONNSP_1:31
for b1 being TopSpace holds
( ex b2 being Point of b1 st
for b3 being Point of b1 holds b2,b3 are_joined iff for b2, b3 being Point of b1 holds b2,b3 are_joined )
proof end;

theorem Th32: :: CONNSP_1:32
for b1 being non empty TopSpace holds
( ( for b2, b3 being Point of b1 holds b2,b3 are_joined ) implies b1 is connected )
proof end;

theorem Th33: :: CONNSP_1:33
for b1 being non empty TopSpace
for b2 being Point of b1
for b3 being Subset-Family of b1 holds
not ( ( for b4 being Subset of b1 holds
( b4 in b3 iff ( b4 is connected & b2 in b4 ) ) ) & not b3 <> {} )
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
pred c2 is_a_component_of c1 means :Def5: :: CONNSP_1:def 5
( a2 is connected & ( for b1 being Subset of a1 holds
( b1 is connected & a2 c= b1 implies a2 = b1 ) ) );
end;

:: deftheorem Def5 defines is_a_component_of CONNSP_1:def 5 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is_a_component_of b1 iff ( b2 is connected & ( for b3 being Subset of b1 holds
( b3 is connected & b2 c= b3 implies b2 = b3 ) ) ) );

theorem Th34: :: CONNSP_1:34
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
not ( b2 is_a_component_of b1 & not b2 <> {} b1 )
proof end;

theorem Th35: :: CONNSP_1:35
for b1 being TopSpace
for b2 being Subset of b1 holds
( b2 is_a_component_of b1 implies b2 is closed )
proof end;

theorem Th36: :: CONNSP_1:36
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
not ( b2 is_a_component_of b1 & b3 is_a_component_of b1 & not b2 = b3 & not b2,b3 are_separated )
proof end;

theorem Th37: :: CONNSP_1:37
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
not ( b2 is_a_component_of b1 & b3 is_a_component_of b1 & not b2 = b3 & not b2 misses b3 )
proof end;

theorem Th38: :: CONNSP_1:38
for b1 being TopSpace
for b2 being Subset of b1 holds
( b2 is connected implies for b3 being Subset of b1 holds
not ( b3 is_a_component_of b1 & not b2 misses b3 & not b2 c= b3 ) )
proof end;

definition
let c1 be TopStruct ;
let c2, c3 be Subset of c1;
pred c3 is_a_component_of c2 means :Def6: :: CONNSP_1:def 6
ex b1 being Subset of (a1 | a2) st
( b1 = a3 & b1 is_a_component_of a1 | a2 );
end;

:: deftheorem Def6 defines is_a_component_of CONNSP_1:def 6 :
for b1 being TopStruct
for b2, b3 being Subset of b1 holds
( b3 is_a_component_of b2 iff ex b4 being Subset of (b1 | b2) st
( b4 = b3 & b4 is_a_component_of b1 | b2 ) );

theorem Th39: :: CONNSP_1:39
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b1 is connected & b2 is connected & b3 is_a_component_of ([#] b1) \ b2 implies ([#] b1) \ b3 is connected )
proof end;

definition
let c1 be TopStruct ;
let c2 be Point of c1;
func skl c2 -> Subset of a1 means :Def7: :: CONNSP_1:def 7
ex b1 being Subset-Family of a1 st
( ( for b2 being Subset of a1 holds
( b2 in b1 iff ( b2 is connected & a2 in b2 ) ) ) & union b1 = a3 );
existence
ex b1 being Subset of c1ex b2 being Subset-Family of c1 st
( ( for b3 being Subset of c1 holds
( b3 in b2 iff ( b3 is connected & c2 in b3 ) ) ) & union b2 = b1 )
proof end;
uniqueness
for b1, b2 being Subset of c1 holds
( ex b3 being Subset-Family of c1 st
( ( for b4 being Subset of c1 holds
( b4 in b3 iff ( b4 is connected & c2 in b4 ) ) ) & union b3 = b1 ) & ex b3 being Subset-Family of c1 st
( ( for b4 being Subset of c1 holds
( b4 in b3 iff ( b4 is connected & c2 in b4 ) ) ) & union b3 = b2 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def7 defines skl CONNSP_1:def 7 :
for b1 being TopStruct
for b2 being Point of b1
for b3 being Subset of b1 holds
( b3 = skl b2 iff ex b4 being Subset-Family of b1 st
( ( for b5 being Subset of b1 holds
( b5 in b4 iff ( b5 is connected & b2 in b5 ) ) ) & union b4 = b3 ) );

theorem Th40: :: CONNSP_1:40
for b1 being non empty TopSpace
for b2 being Point of b1 holds b2 in skl b2
proof end;

theorem Th41: :: CONNSP_1:41
for b1 being non empty TopSpace
for b2 being Point of b1 holds skl b2 is connected
proof end;

theorem Th42: :: CONNSP_1:42
for b1 being non empty TopSpace
for b2 being Subset of b1
for b3 being Point of b1 holds
( b2 is connected & skl b3 c= b2 implies b2 = skl b3 )
proof end;

theorem Th43: :: CONNSP_1:43
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is_a_component_of b1 iff ex b3 being Point of b1 st b2 = skl b3 )
proof end;

theorem Th44: :: CONNSP_1:44
for b1 being non empty TopSpace
for b2 being Subset of b1
for b3 being Point of b1 holds
( b2 is_a_component_of b1 & b3 in b2 implies b2 = skl b3 )
proof end;

theorem Th45: :: CONNSP_1:45
for b1 being non empty TopSpace
for b2 being Subset of b1
for b3 being Point of b1 holds
( b2 = skl b3 implies for b4 being Point of b1 holds
( b4 in b2 implies skl b4 = b2 ) )
proof end;

theorem Th46: :: CONNSP_1:46
for b1 being non empty TopSpace
for b2 being Subset-Family of b1 holds
( ( for b3 being Subset of b1 holds
( b3 in b2 iff b3 is_a_component_of b1 ) ) implies b2 is_a_cover_of b1 )
proof end;