Article CONNSP_1, MML version 4.99.1005
:: CONNSP_1:prednot 1 => CONNSP_1:pred 1
definition
let a1 be TopStruct;
let a2, a3 be Element of bool the carrier of a1;
pred A2,A3 are_separated means
Cl a2 misses a3 & a2 misses Cl a3;
symmetry;
:: for a1 being TopStruct
:: for a2, a3 being Element of bool the carrier of a1
:: st a2,a3 are_separated
:: holds a3,a2 are_separated;
end;
:: CONNSP_1:dfs 1
definiens
let a1 be TopStruct;
let a2, a3 be Element of bool the carrier of a1;
To prove
a2,a3 are_separated
it is sufficient to prove
thus Cl a2 misses a3 & a2 misses Cl a3;
:: CONNSP_1:def 1
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2,b3 are_separated
iff
Cl b2 misses b3 & b2 misses Cl b3;
:: CONNSP_1:th 2
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 are_separated
holds b2 misses b3;
:: CONNSP_1:th 3
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st [#] b1 = b2 \/ b3 & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
holds b2,b3 are_separated;
:: CONNSP_1:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st [#] b1 = b2 \/ b3 & b2 is open(b1) & b3 is open(b1) & b2 misses b3
holds b2,b3 are_separated;
:: CONNSP_1:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st [#] b1 = b2 \/ b3 & b2,b3 are_separated
holds b2 is open(b1) & b2 is closed(b1) & b3 is open(b1) & b3 is closed(b1);
:: CONNSP_1:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5, b6 being Element of bool the carrier of b2
st b5 = b3 & b6 = b4 & b5,b6 are_separated
holds b3,b4 are_separated;
:: CONNSP_1:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5, b6 being Element of bool the carrier of b2
st b3 = b5 & b4 = b6 & b3 \/ b4 c= [#] b2 & b3,b4 are_separated
holds b5,b6 are_separated;
:: CONNSP_1:th 8
theorem
for b1 being TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
st b2,b3 are_separated & b4 c= b2 & b5 c= b3
holds b4,b5 are_separated;
:: CONNSP_1:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2,b3 are_separated & b2,b4 are_separated
holds b2,b3 \/ b4 are_separated;
:: CONNSP_1:th 10
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st (b2 is closed(b1) & b3 is closed(b1) or b2 is open(b1) & b3 is open(b1))
holds b2 \ b3,b3 \ b2 are_separated;
:: CONNSP_1:attrnot 1 => CONNSP_1:attr 1
definition
let a1 be TopStruct;
attr a1 is connected means
for b1, b2 being Element of bool the carrier of a1
st [#] a1 = b1 \/ b2 & b1,b2 are_separated & b1 <> {} a1
holds b2 = {} a1;
end;
:: CONNSP_1:dfs 2
definiens
let a1 be TopStruct;
To prove
a1 is connected
it is sufficient to prove
thus for b1, b2 being Element of bool the carrier of a1
st [#] a1 = b1 \/ b2 & b1,b2 are_separated & b1 <> {} a1
holds b2 = {} a1;
:: CONNSP_1:def 2
theorem
for b1 being TopStruct holds
b1 is connected
iff
for b2, b3 being Element of bool the carrier of b1
st [#] b1 = b2 \/ b3 & b2,b3 are_separated & b2 <> {} b1
holds b3 = {} b1;
:: CONNSP_1:th 11
theorem
for b1 being TopSpace-like TopStruct holds
b1 is connected
iff
for b2, b3 being Element of bool the carrier of b1
st [#] b1 = b2 \/ b3 & b2 <> {} b1 & b3 <> {} b1 & b2 is closed(b1) & b3 is closed(b1)
holds b2 meets b3;
:: CONNSP_1:th 12
theorem
for b1 being TopSpace-like TopStruct holds
b1 is connected
iff
for b2, b3 being Element of bool the carrier of b1
st [#] b1 = b2 \/ b3 & b2 <> {} b1 & b3 <> {} b1 & b2 is open(b1) & b3 is open(b1)
holds b2 meets b3;
:: CONNSP_1:th 13
theorem
for b1 being TopSpace-like TopStruct holds
b1 is connected
iff
for b2 being Element of bool the carrier of b1
st b2 <> {} b1 & b2 <> [#] b1
holds Cl b2 meets Cl (([#] b1) \ b2);
:: CONNSP_1:th 14
theorem
for b1 being TopSpace-like TopStruct holds
b1 is connected
iff
for b2 being Element of bool the carrier of b1
st b2 is open(b1) & b2 is closed(b1) & b2 <> {} b1
holds b2 = [#] b1;
:: CONNSP_1:th 15
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is continuous(b1, b2) & b3 .: [#] b1 = [#] b2 & b1 is connected
holds b2 is connected;
:: CONNSP_1:attrnot 2 => CONNSP_1:attr 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is connected means
a1 | a2 is connected;
end;
:: CONNSP_1:dfs 3
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is connected
it is sufficient to prove
thus a1 | a2 is connected;
:: CONNSP_1:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is connected(b1)
iff
b1 | b2 is connected;
:: CONNSP_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is connected(b1)
iff
for b3, b4 being Element of bool the carrier of b1
st b2 = b3 \/ b4 & b3,b4 are_separated & b3 <> {} b1
holds b4 = {} b1;
:: CONNSP_1:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 c= b3 \/ b4 & b3,b4 are_separated & not b2 c= b3
holds b2 c= b4;
:: CONNSP_1:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b3 is connected(b1) & not b2,b3 are_separated
holds b2 \/ b3 is connected(b1);
:: CONNSP_1:th 19
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 c= b3 & b3 c= Cl b2
holds b3 is connected(b1);
:: CONNSP_1:th 20
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is connected(b1)
holds Cl b2 is connected(b1);
:: CONNSP_1:th 21
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b1 is connected & b2 is connected(b1) & ([#] b1) \ b2 = b3 \/ b4 & b3,b4 are_separated
holds b2 \/ b3 is connected(b1) & b2 \/ b4 is connected(b1);
:: CONNSP_1:th 22
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st ([#] b1) \ b2 = b3 \/ b4 & b3,b4 are_separated & b2 is closed(b1)
holds b2 \/ b3 is closed(b1) & b2 \/ b4 is closed(b1);
:: CONNSP_1:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is connected(b1) & b2 meets b3 & b2 \ b3 <> {} b1
holds b2 meets Fr b3;
:: CONNSP_1:th 24
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds b3 is connected(b1)
iff
b4 is connected(b2);
:: CONNSP_1:th 25
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is closed(b1) & b3 is closed(b1) & b2 \/ b3 is connected(b1) & b2 /\ b3 is connected(b1)
holds b2 is connected(b1) & b3 is connected(b1);
:: CONNSP_1:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st (for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is connected(b1)) &
(ex b3 being Element of bool the carrier of b1 st
b3 <> {} b1 &
b3 in b2 &
(for b4 being Element of bool the carrier of b1
st b4 in b2 & b4 <> b3
holds not b3,b4 are_separated))
holds union b2 is connected(b1);
:: CONNSP_1:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st (for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is connected(b1)) &
meet b2 <> {} b1
holds union b2 is connected(b1);
:: CONNSP_1:th 28
theorem
for b1 being TopSpace-like TopStruct holds
[#] b1 is connected(b1)
iff
b1 is connected;
:: CONNSP_1:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
{b2} is connected(b1);
:: CONNSP_1:prednot 2 => CONNSP_1:pred 2
definition
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
pred A2,A3 are_joined means
ex b1 being Element of bool the carrier of a1 st
b1 is connected(a1) & a2 in b1 & a3 in b1;
end;
:: CONNSP_1:dfs 4
definiens
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
To prove
a2,a3 are_joined
it is sufficient to prove
thus ex b1 being Element of bool the carrier of a1 st
b1 is connected(a1) & a2 in b1 & a3 in b1;
:: CONNSP_1:def 4
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_joined
iff
ex b4 being Element of bool the carrier of b1 st
b4 is connected(b1) & b2 in b4 & b3 in b4;
:: CONNSP_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
st ex b2 being Element of the carrier of b1 st
for b3 being Element of the carrier of b1 holds
b2,b3 are_joined
holds b1 is connected;
:: CONNSP_1:th 31
theorem
for b1 being TopSpace-like TopStruct holds
ex b2 being Element of the carrier of b1 st
for b3 being Element of the carrier of b1 holds
b2,b3 are_joined
iff
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_joined;
:: CONNSP_1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_joined
holds b1 is connected;
:: CONNSP_1:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool bool the carrier of b1
st for b4 being Element of bool the carrier of b1 holds
b4 in b3
iff
b4 is connected(b1) & b2 in b4
holds b3 <> {};
:: CONNSP_1:prednot 3 => CONNSP_1:pred 3
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
pred A2 is_a_component_of A1 means
a2 is connected(a1) &
(for b1 being Element of bool the carrier of a1
st b1 is connected(a1) & a2 c= b1
holds a2 = b1);
end;
:: CONNSP_1:dfs 5
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is_a_component_of a1
it is sufficient to prove
thus a2 is connected(a1) &
(for b1 being Element of bool the carrier of a1
st b1 is connected(a1) & a2 c= b1
holds a2 = b1);
:: CONNSP_1:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is_a_component_of b1
iff
b2 is connected(b1) &
(for b3 being Element of bool the carrier of b1
st b3 is connected(b1) & b2 c= b3
holds b2 = b3);
:: CONNSP_1:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is_a_component_of b1
holds b2 <> {} b1;
:: CONNSP_1:th 35
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is_a_component_of b1
holds b2 is closed(b1);
:: CONNSP_1:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is_a_component_of b1 & b3 is_a_component_of b1 & b2 <> b3
holds b2,b3 are_separated;
:: CONNSP_1:th 37
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is_a_component_of b1 & b3 is_a_component_of b1 & b2 <> b3
holds b2 misses b3;
:: CONNSP_1:th 38
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is connected(b1)
for b3 being Element of bool the carrier of b1
st b3 is_a_component_of b1 & b2 meets b3
holds b2 c= b3;
:: CONNSP_1:prednot 4 => CONNSP_1:pred 4
definition
let a1 be TopStruct;
let a2, a3 be Element of bool the carrier of a1;
pred A3 is_a_component_of A2 means
ex b1 being Element of bool the carrier of a1 | a2 st
b1 = a3 & b1 is_a_component_of a1 | a2;
end;
:: CONNSP_1:dfs 6
definiens
let a1 be TopStruct;
let a2, a3 be Element of bool the carrier of a1;
To prove
a3 is_a_component_of a2
it is sufficient to prove
thus ex b1 being Element of bool the carrier of a1 | a2 st
b1 = a3 & b1 is_a_component_of a1 | a2;
:: CONNSP_1:def 6
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b3 is_a_component_of b2
iff
ex b4 being Element of bool the carrier of b1 | b2 st
b4 = b3 & b4 is_a_component_of b1 | b2;
:: CONNSP_1:th 39
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b1 is connected & b2 is connected(b1) & b3 is_a_component_of ([#] b1) \ b2
holds ([#] b1) \ b3 is connected(b1);
:: CONNSP_1:funcnot 1 => CONNSP_1:func 1
definition
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
func Component_of A2 -> Element of bool the carrier of a1 means
ex b1 being Element of bool bool the carrier of a1 st
(for b2 being Element of bool the carrier of a1 holds
b2 in b1
iff
b2 is connected(a1) & a2 in b2) &
union b1 = it;
end;
:: CONNSP_1:def 7
theorem
for b1 being TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 = Component_of b2
iff
ex b4 being Element of bool bool the carrier of b1 st
(for b5 being Element of bool the carrier of b1 holds
b5 in b4
iff
b5 is connected(b1) & b2 in b5) &
union b4 = b3;
:: CONNSP_1:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
b2 in Component_of b2;
:: CONNSP_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
Component_of b2 is connected(b1);
:: CONNSP_1:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 is connected(b1) & Component_of b3 c= b2
holds b2 = Component_of b3;
:: CONNSP_1:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is_a_component_of b1
iff
ex b3 being Element of the carrier of b1 st
b2 = Component_of b3;
:: CONNSP_1:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 is_a_component_of b1 & b3 in b2
holds b2 = Component_of b3;
:: CONNSP_1:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 = Component_of b3
for b4 being Element of the carrier of b1
st b4 in b2
holds Component_of b4 = b2;
:: CONNSP_1:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1 holds
b3 in b2
iff
b3 is_a_component_of b1
holds b2 is_a_cover_of b1;