Article TOPS_1, MML version 4.99.1005
:: TOPS_1:th 21
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool the carrier of b1
st b2 ` = b3 `
holds b2 = b3;
:: TOPS_1:th 22
theorem
for b1 being TopSpace-like TopStruct holds
{} b1 is closed(b1);
:: TOPS_1:funcreg 1
registration
let a1 be TopSpace-like TopStruct;
cluster {} a1 -> closed;
end;
:: TOPS_1:th 26
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl Cl b2 = Cl b2;
:: TOPS_1:th 27
theorem
for b1 being TopStruct holds
Cl [#] b1 = [#] b1;
:: TOPS_1:funcreg 2
registration
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
cluster Cl a2 -> closed;
end;
:: TOPS_1:th 29
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
b2 ` is open(b1);
:: TOPS_1:funcreg 3
registration
let a1 be TopSpace-like TopStruct;
let a2 be closed Element of bool the carrier of a1;
cluster a2 ` -> open;
end;
:: TOPS_1:th 30
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
b2 ` is closed(b1);
:: TOPS_1:exreg 1
registration
let a1 be TopSpace-like TopStruct;
cluster open Element of bool the carrier of a1;
end;
:: TOPS_1:funcreg 4
registration
let a1 be TopSpace-like TopStruct;
let a2 be open Element of bool the carrier of a1;
cluster a2 ` -> closed;
end;
:: TOPS_1:th 31
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is closed(b1) & b3 c= b2
holds Cl b3 c= b2;
:: TOPS_1:th 32
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Cl b2) \ Cl b3 c= Cl (b2 \ b3);
:: TOPS_1:th 34
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)
holds Cl (b2 /\ b3) = (Cl b2) /\ Cl b3;
:: TOPS_1:th 35
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)
holds b2 /\ b3 is closed(b1);
:: TOPS_1:th 36
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)
holds b2 \/ b3 is closed(b1);
:: TOPS_1:th 37
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1) & b3 is open(b1)
holds b2 \/ b3 is open(b1);
:: TOPS_1:th 38
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1) & b3 is open(b1)
holds b2 /\ b3 is open(b1);
:: TOPS_1:th 39
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 holds
b3 in Cl b2
iff
for b4 being Element of bool the carrier of b1
st b4 is open(b1) & b3 in b4
holds b2 meets b4;
:: TOPS_1:th 40
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1)
holds b2 /\ Cl b3 c= Cl (b2 /\ b3);
:: TOPS_1:th 41
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1)
holds Cl (b2 /\ Cl b3) = Cl (b2 /\ b3);
:: TOPS_1:funcnot 1 => TOPS_1:func 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
func Int A2 -> Element of bool the carrier of a1 equals
(Cl (a2 `)) `;
projectivity;
:: for a1 being TopStruct
:: for a2 being Element of bool the carrier of a1 holds
:: Int Int a2 = Int a2;
end;
:: TOPS_1:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Int b2 = (Cl (b2 `)) `;
:: TOPS_1:th 43
theorem
for b1 being TopSpace-like TopStruct holds
Int [#] b1 = [#] b1;
:: TOPS_1:th 44
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Int b2 c= b2;
:: TOPS_1:th 45
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Int Int b2 = Int b2;
:: TOPS_1:th 46
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int b2) /\ Int b3 = Int (b2 /\ b3);
:: TOPS_1:th 47
theorem
for b1 being TopStruct holds
Int {} b1 = {} b1;
:: TOPS_1:funcreg 5
registration
let a1 be TopStruct;
cluster Int {} a1 -> empty;
end;
:: TOPS_1:th 48
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds Int b2 c= Int b3;
:: TOPS_1:th 49
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int b2) \/ Int b3 c= Int (b2 \/ b3);
:: TOPS_1:th 50
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Int (b2 \ b3) c= (Int b2) \ Int b3;
:: TOPS_1:th 51
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Int b2 is open(b1);
:: TOPS_1:funcreg 6
registration
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
cluster Int a2 -> open;
end;
:: TOPS_1:th 52
theorem
for b1 being TopSpace-like TopStruct holds
{} b1 is open(b1);
:: TOPS_1:th 53
theorem
for b1 being TopSpace-like TopStruct holds
[#] b1 is open(b1);
:: TOPS_1:funcreg 7
registration
let a1 be TopSpace-like TopStruct;
cluster {} a1 -> open;
end;
:: TOPS_1:funcreg 8
registration
let a1 be TopSpace-like TopStruct;
cluster [#] a1 -> open;
end;
:: TOPS_1:condreg 1
registration
let a1 be TopSpace-like TopStruct;
cluster empty -> open closed (Element of bool the carrier of a1);
end;
:: TOPS_1:exreg 2
registration
let a1 be TopSpace-like TopStruct;
cluster open closed Element of bool the carrier of a1;
end;
:: TOPS_1:exreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty open closed Element of bool the carrier of a1;
end;
:: TOPS_1:th 54
theorem
for b1 being TopSpace-like TopStruct
for b2 being set
for b3 being Element of bool the carrier of b1 holds
b2 in Int b3
iff
ex b4 being Element of bool the carrier of b1 st
b4 is open(b1) & b4 c= b3 & b2 in b4;
:: TOPS_1:th 55
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
(b4 is open(b2) implies Int b4 = b4) & (Int b3 = b3 implies b3 is open(b1));
:: TOPS_1:th 56
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1) & b2 c= b3
holds b2 c= Int b3;
:: TOPS_1:th 57
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
for b3 being set holds
b3 in b2
iff
ex b4 being Element of bool the carrier of b1 st
b4 is open(b1) & b4 c= b2 & b3 in b4;
:: TOPS_1:th 58
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl Int b2 = Cl Int Cl Int b2;
:: TOPS_1:th 59
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is open(b1)
holds Cl Int Cl b2 = Cl b2;
:: TOPS_1:funcnot 2 => TOPS_1:func 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
func Fr A2 -> Element of bool the carrier of a1 equals
(Cl a2) /\ Cl (a2 `);
end;
:: TOPS_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr b2 = (Cl b2) /\ Cl (b2 `);
:: TOPS_1:th 60
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr b2 is closed(b1);
:: TOPS_1:funcreg 9
registration
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
cluster Fr a2 -> closed;
end;
:: TOPS_1:th 61
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 holds
b3 in Fr b2
iff
for b4 being Element of bool the carrier of b1
st b4 is open(b1) & b3 in b4
holds b2 meets b4 & b2 ` meets b4;
:: TOPS_1:th 62
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr b2 = Fr (b2 `);
:: TOPS_1:th 64
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr b2 = ((Cl (b2 `)) /\ b2) \/ ((Cl b2) \ b2);
:: TOPS_1:th 65
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl b2 = b2 \/ Fr b2;
:: TOPS_1:th 66
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Fr (b2 /\ b3) c= (Fr b2) \/ Fr b3;
:: TOPS_1:th 67
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Fr (b2 \/ b3) c= (Fr b2) \/ Fr b3;
:: TOPS_1:th 68
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr Fr b2 c= Fr b2;
:: TOPS_1:th 69
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
holds Fr b2 c= b2;
:: TOPS_1:th 70
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Fr b2) \/ Fr b3 = ((Fr (b2 \/ b3)) \/ Fr (b2 /\ b3)) \/ ((Fr b2) /\ Fr b3);
:: TOPS_1:th 71
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr Int b2 c= Fr b2;
:: TOPS_1:th 72
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr Cl b2 c= Fr b2;
:: TOPS_1:th 73
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Int b2 misses Fr b2;
:: TOPS_1:th 74
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Int b2 = b2 \ Fr b2;
:: TOPS_1:th 75
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr Fr Fr b2 = Fr Fr b2;
:: TOPS_1:th 76
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
Fr b2 = (Cl b2) \ b2;
:: TOPS_1:th 77
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
Fr b2 = b2 \ Int b2;
:: TOPS_1:attrnot 1 => TOPS_1:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is dense means
Cl a2 = [#] a1;
end;
:: TOPS_1:dfs 3
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is dense
it is sufficient to prove
thus Cl a2 = [#] a1;
:: TOPS_1:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is dense(b1)
iff
Cl b2 = [#] b1;
:: TOPS_1:funcreg 10
registration
let a1 be TopStruct;
cluster [#] a1 -> dense;
end;
:: TOPS_1:th 79
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is dense(b1) & b2 c= b3
holds b3 is dense(b1);
:: TOPS_1:th 80
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is dense(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 <> {} & b3 is open(b1)
holds b2 meets b3;
:: TOPS_1:th 81
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is dense(b1)
for b3 being Element of bool the carrier of b1
st b3 is open(b1)
holds Cl b3 = Cl (b3 /\ b2);
:: TOPS_1:th 82
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is dense(b1) & b3 is dense(b1) & b3 is open(b1)
holds b2 /\ b3 is dense(b1);
:: TOPS_1:attrnot 2 => TOPS_1:attr 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is boundary means
a2 ` is dense(a1);
end;
:: TOPS_1:dfs 4
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is boundary
it is sufficient to prove
thus a2 ` is dense(a1);
:: TOPS_1:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is boundary(b1)
iff
b2 ` is dense(b1);
:: TOPS_1:condreg 2
registration
let a1 be TopStruct;
cluster empty -> boundary (Element of bool the carrier of a1);
end;
:: TOPS_1:exreg 4
registration
let a1 be TopStruct;
cluster empty Element of bool the carrier of a1;
end;
:: TOPS_1:th 84
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is boundary(b1)
iff
Int b2 = {};
:: TOPS_1:funcreg 11
registration
let a1 be TopStruct;
let a2 be boundary Element of bool the carrier of a1;
cluster Int a2 -> empty;
end;
:: TOPS_1:th 85
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is boundary(b1) & b3 is boundary(b1) & b3 is closed(b1)
holds b2 \/ b3 is boundary(b1);
:: TOPS_1:th 86
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is boundary(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 c= b2 & b3 is open(b1)
holds b3 = {};
:: TOPS_1:th 87
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
holds b2 is boundary(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 <> {} & b3 is open(b1)
holds ex b4 being Element of bool the carrier of b1 st
b4 c= b3 & b4 <> {} & b4 is open(b1) & b2 misses b4;
:: TOPS_1:th 88
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is boundary(b1)
iff
b2 c= Fr b2;
:: TOPS_1:attrnot 3 => TOPS_1:attr 3
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is nowhere_dense means
Cl a2 is boundary(a1);
end;
:: TOPS_1:dfs 5
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is nowhere_dense
it is sufficient to prove
thus Cl a2 is boundary(a1);
:: TOPS_1:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is nowhere_dense(b1)
iff
Cl b2 is boundary(b1);
:: TOPS_1:condreg 3
registration
let a1 be TopSpace-like TopStruct;
cluster empty -> nowhere_dense (Element of bool the carrier of a1);
end;
:: TOPS_1:exreg 5
registration
let a1 be TopSpace-like TopStruct;
cluster empty Element of bool the carrier of a1;
end;
:: TOPS_1:th 90
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is nowhere_dense(b1) & b3 is nowhere_dense(b1)
holds b2 \/ b3 is nowhere_dense(b1);
:: TOPS_1:th 91
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is nowhere_dense(b1)
holds b2 ` is dense(b1);
:: TOPS_1:funcreg 12
registration
let a1 be TopSpace-like TopStruct;
let a2 be nowhere_dense Element of bool the carrier of a1;
cluster a2 ` -> dense;
end;
:: TOPS_1:th 92
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is nowhere_dense(b1)
holds b2 is boundary(b1);
:: TOPS_1:condreg 4
registration
let a1 be TopSpace-like TopStruct;
cluster nowhere_dense -> boundary (Element of bool the carrier of a1);
end;
:: TOPS_1:th 93
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is boundary(b1) & b2 is closed(b1)
holds b2 is nowhere_dense(b1);
:: TOPS_1:condreg 5
registration
let a1 be TopSpace-like TopStruct;
cluster closed boundary -> nowhere_dense (Element of bool the carrier of a1);
end;
:: TOPS_1:th 94
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
holds b2 is nowhere_dense(b1)
iff
b2 = Fr b2;
:: TOPS_1:th 95
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is open(b1)
holds Fr b2 is nowhere_dense(b1);
:: TOPS_1:funcreg 13
registration
let a1 be TopSpace-like TopStruct;
let a2 be open Element of bool the carrier of a1;
cluster Fr a2 -> nowhere_dense;
end;
:: TOPS_1:th 96
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
holds Fr b2 is nowhere_dense(b1);
:: TOPS_1:funcreg 14
registration
let a1 be TopSpace-like TopStruct;
let a2 be closed Element of bool the carrier of a1;
cluster Fr a2 -> nowhere_dense;
end;
:: TOPS_1:th 97
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is open(b1) & b2 is nowhere_dense(b1)
holds b2 = {};
:: TOPS_1:condreg 6
registration
let a1 be TopSpace-like TopStruct;
cluster open nowhere_dense -> empty (Element of bool the carrier of a1);
end;
:: TOPS_1:attrnot 4 => TOPS_1:attr 4
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is condensed means
Int Cl a2 c= a2 & a2 c= Cl Int a2;
end;
:: TOPS_1:dfs 6
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is condensed
it is sufficient to prove
thus Int Cl a2 c= a2 & a2 c= Cl Int a2;
:: TOPS_1:def 6
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is condensed(b1)
iff
Int Cl b2 c= b2 & b2 c= Cl Int b2;
:: TOPS_1:attrnot 5 => TOPS_1:attr 5
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is closed_condensed means
a2 = Cl Int a2;
end;
:: TOPS_1:dfs 7
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is closed_condensed
it is sufficient to prove
thus a2 = Cl Int a2;
:: TOPS_1:def 7
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed_condensed(b1)
iff
b2 = Cl Int b2;
:: TOPS_1:attrnot 6 => TOPS_1:attr 6
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is open_condensed means
a2 = Int Cl a2;
end;
:: TOPS_1:dfs 8
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is open_condensed
it is sufficient to prove
thus a2 = Int Cl a2;
:: TOPS_1:def 8
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open_condensed(b1)
iff
b2 = Int Cl b2;
:: TOPS_1:th 101
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open_condensed(b1)
iff
b2 ` is closed_condensed(b1);
:: TOPS_1:th 102
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed_condensed(b1)
holds Fr Int b2 = Fr b2;
:: TOPS_1:th 103
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed_condensed(b1)
holds Fr b2 c= Cl Int b2;
:: TOPS_1:th 104
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is open_condensed(b1)
holds Fr b2 = Fr Cl b2 & Fr Cl b2 = (Cl b2) \ b2;
:: TOPS_1:th 105
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is open(b1) & b2 is closed(b1)
holds b2 is closed_condensed(b1)
iff
b2 is open_condensed(b1);
:: TOPS_1:th 106
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
(b4 is closed(b2) & b4 is condensed(b2) implies b4 is closed_condensed(b2)) &
(b3 is closed_condensed(b1) implies b3 is closed(b1) & b3 is condensed(b1));
:: TOPS_1:th 107
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
(b4 is open(b2) & b4 is condensed(b2) implies b4 is open_condensed(b2)) & (b3 is open_condensed(b1) implies b3 is open(b1) & b3 is condensed(b1));
:: TOPS_1:th 108
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is closed_condensed(b1) & b3 is closed_condensed(b1)
holds b2 \/ b3 is closed_condensed(b1);
:: TOPS_1:th 109
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open_condensed(b1) & b3 is open_condensed(b1)
holds b2 /\ b3 is open_condensed(b1);
:: TOPS_1:th 110
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is condensed(b1)
holds Int Fr b2 = {};
:: TOPS_1:th 111
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is condensed(b1)
holds Int b2 is condensed(b1) & Cl b2 is condensed(b1);