Article TOPGEN_1, MML version 4.99.1005
:: TOPGEN_1:th 1
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool the carrier of b1 holds
b2 meets b3 `
iff
b2 \ b3 <> {};
:: TOPGEN_1:th 3
theorem
for b1 being 1-sorted holds
b1 is countable
iff
Card [#] b1 c= alef 0;
:: TOPGEN_1:funcreg 1
registration
let a1 be finite 1-sorted;
cluster [#] a1 -> finite;
end;
:: TOPGEN_1:condreg 1
registration
cluster finite -> countable (1-sorted);
end;
:: TOPGEN_1:exreg 1
registration
cluster non empty countable 1-sorted;
end;
:: TOPGEN_1:exreg 2
registration
cluster non empty TopSpace-like countable TopStruct;
end;
:: TOPGEN_1:funcreg 2
registration
let a1 be countable 1-sorted;
cluster [#] a1 -> countable;
end;
:: TOPGEN_1:exreg 3
registration
cluster non empty TopSpace-like being_T1 TopStruct;
end;
:: TOPGEN_1:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Int b2 = (Cl (b2 `)) `;
:: TOPGEN_1:funcnot 1 => TOPGEN_1:func 1
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
func Fr A2 -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1 holds
b1 in it
iff
ex b2 being Element of bool the carrier of a1 st
b1 = Fr b2 & b2 in a2;
end;
:: TOPGEN_1:def 1
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
b3 = Fr b2
iff
for b4 being Element of bool the carrier of b1 holds
b4 in b3
iff
ex b5 being Element of bool the carrier of b1 st
b4 = Fr b5 & b5 in b2;
:: TOPGEN_1:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 = {}
holds Fr b2 = {};
:: TOPGEN_1:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1
st b2 = {b3}
holds Fr b2 = {Fr b3};
:: TOPGEN_1:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 c= b3
holds Fr b2 c= Fr b3;
:: TOPGEN_1:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
Fr (b2 \/ b3) = (Fr b2) \/ Fr b3;
:: TOPGEN_1:th 10
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
Fr b2 = (Cl b2) \ Int b2;
:: TOPGEN_1:th 11
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 & b4 \ b2 <> {};
:: TOPGEN_1:th 12
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 Basis of b3
for b5 being Element of bool the carrier of b1
st b5 in b4
holds b2 meets b5 & b5 \ b2 <> {};
:: TOPGEN_1:th 13
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
ex b4 being Basis of b3 st
for b5 being Element of bool the carrier of b1
st b5 in b4
holds b2 meets b5 & b5 \ b2 <> {};
:: TOPGEN_1:th 14
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Fr (b2 /\ b3) c= ((Cl b2) /\ Fr b3) \/ ((Fr b2) /\ Cl b3);
:: TOPGEN_1:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
the carrier of b1 = ((Int b2) \/ Fr b2) \/ Int (b2 `);
:: TOPGEN_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1) & b2 is closed(b1)
iff
Fr b2 = {};
:: TOPGEN_1:prednot 1 => TOPGEN_1:pred 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be set;
pred A3 is_an_accumulation_point_of A2 means
a3 in Cl (a2 \ {a3});
end;
:: TOPGEN_1:dfs 2
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be set;
To prove
a3 is_an_accumulation_point_of a2
it is sufficient to prove
thus a3 in Cl (a2 \ {a3});
:: TOPGEN_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
b3 is_an_accumulation_point_of b2
iff
b3 in Cl (b2 \ {b3});
:: TOPGEN_1:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set
st b3 is_an_accumulation_point_of b2
holds b3 is Element of the carrier of b1;
:: TOPGEN_1:funcnot 2 => TOPGEN_1:func 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
func Der A2 -> Element of bool the carrier of a1 means
for b1 being set
st b1 in the carrier of a1
holds b1 in it
iff
b1 is_an_accumulation_point_of a2;
end;
:: TOPGEN_1:def 3
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b3 = Der b2
iff
for b4 being set
st b4 in the carrier of b1
holds b4 in b3
iff
b4 is_an_accumulation_point_of b2;
:: TOPGEN_1:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
b3 in Der b2
iff
b3 is_an_accumulation_point_of b2;
:: TOPGEN_1:th 19
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 Der b2
iff
for b4 being open Element of bool the carrier of b1
st b3 in b4
holds ex b5 being Element of the carrier of b1 st
b5 in b2 /\ b4 & b3 <> b5;
:: TOPGEN_1:th 20
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 Der b2
iff
for b4 being Basis of b3
for b5 being Element of bool the carrier of b1
st b5 in b4
holds ex b6 being Element of the carrier of b1 st
b6 in b2 /\ b5 & b3 <> b6;
:: TOPGEN_1:th 21
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 Der b2
iff
ex b4 being Basis of b3 st
for b5 being Element of bool the carrier of b1
st b5 in b4
holds ex b6 being Element of the carrier of b1 st
b6 in b2 /\ b5 & b3 <> b6;
:: TOPGEN_1:prednot 2 => TOPGEN_1:pred 2
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be set;
pred A3 is_isolated_in A2 means
a3 in a2 & not a3 is_an_accumulation_point_of a2;
end;
:: TOPGEN_1:dfs 4
definiens
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be set;
To prove
a3 is_isolated_in a2
it is sufficient to prove
thus a3 in a2 & not a3 is_an_accumulation_point_of a2;
:: TOPGEN_1:def 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
b3 is_isolated_in b2
iff
b3 in b2 & not b3 is_an_accumulation_point_of b2;
:: TOPGEN_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being set holds
b3 in b2 \ Der b2
iff
b3 is_isolated_in b2;
:: TOPGEN_1:th 23
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 is_an_accumulation_point_of b2
iff
for b4 being open Element of bool the carrier of b1
st b3 in b4
holds ex b5 being Element of the carrier of b1 st
b5 <> b3 & b5 in b2 & b5 in b4;
:: TOPGEN_1:th 24
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 is_isolated_in b2
iff
ex b4 being open Element of bool the carrier of b1 st
b4 /\ b2 = {b3};
:: TOPGEN_1:attrnot 1 => TOPGEN_1:attr 1
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
attr a2 is isolated means
a2 is_isolated_in [#] a1;
end;
:: TOPGEN_1:dfs 5
definiens
let a1 be TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
To prove
a2 is isolated
it is sufficient to prove
thus a2 is_isolated_in [#] a1;
:: TOPGEN_1:def 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
b2 is isolated(b1)
iff
b2 is_isolated_in [#] b1;
:: TOPGEN_1:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
b2 is isolated(b1)
iff
{b2} is open(b1);
:: TOPGEN_1:funcnot 3 => TOPGEN_1:func 3
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
func Der A2 -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1 holds
b1 in it
iff
ex b2 being Element of bool the carrier of a1 st
b1 = Der b2 & b2 in a2;
end;
:: TOPGEN_1:def 6
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
b3 = Der b2
iff
for b4 being Element of bool the carrier of b1 holds
b4 in b3
iff
ex b5 being Element of bool the carrier of b1 st
b4 = Der b5 & b5 in b2;
:: TOPGEN_1:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 = {}
holds Der b2 = {};
:: TOPGEN_1:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b3 = {b2}
holds Der b3 = {Der b2};
:: TOPGEN_1:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 c= b3
holds Der b2 c= Der b3;
:: TOPGEN_1:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
Der (b2 \/ b3) = (Der b2) \/ Der b3;
:: TOPGEN_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Der b2 c= Cl b2;
:: TOPGEN_1:th 31
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl b2 = b2 \/ Der b2;
:: TOPGEN_1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds Der b2 c= Der b3;
:: TOPGEN_1:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
Der (b2 \/ b3) = (Der b2) \/ Der b3;
:: TOPGEN_1:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b1 is being_T1
holds Der Der b2 c= Der b2;
:: TOPGEN_1:th 35
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b1 is being_T1
holds Cl Der b2 = Der b2;
:: TOPGEN_1:funcreg 3
registration
let a1 be non empty TopSpace-like being_T1 TopStruct;
let a2 be Element of bool the carrier of a1;
cluster Der a2 -> closed;
end;
:: TOPGEN_1:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
union Der b2 c= Der union b2;
:: TOPGEN_1:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being set
st b2 c= b3 & b4 is_an_accumulation_point_of b2
holds b4 is_an_accumulation_point_of b3;
:: TOPGEN_1:attrnot 2 => TOPGEN_1:attr 2
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is dense-in-itself means
a2 c= Der a2;
end;
:: TOPGEN_1:dfs 7
definiens
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is dense-in-itself
it is sufficient to prove
thus a2 c= Der a2;
:: TOPGEN_1:def 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is dense-in-itself(b1)
iff
b2 c= Der b2;
:: TOPGEN_1:attrnot 3 => TOPGEN_1:attr 3
definition
let a1 be non empty TopSpace-like TopStruct;
attr a1 is dense-in-itself means
[#] a1 is dense-in-itself(a1);
end;
:: TOPGEN_1:dfs 8
definiens
let a1 be non empty TopSpace-like TopStruct;
To prove
a1 is dense-in-itself
it is sufficient to prove
thus [#] a1 is dense-in-itself(a1);
:: TOPGEN_1:def 8
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is dense-in-itself
iff
[#] b1 is dense-in-itself(b1);
:: TOPGEN_1:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b1 is being_T1 & b2 is dense-in-itself(b1)
holds Cl b2 is dense-in-itself(b1);
:: TOPGEN_1:attrnot 4 => TOPGEN_1:attr 4
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is dense-in-itself means
for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is dense-in-itself(a1);
end;
:: TOPGEN_1:dfs 9
definiens
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is dense-in-itself
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is dense-in-itself(a1);
:: TOPGEN_1:def 9
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is dense-in-itself(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is dense-in-itself(b1);
:: TOPGEN_1:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is dense-in-itself(b1)
holds union b2 c= union Der b2;
:: TOPGEN_1:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is dense-in-itself(b1)
holds union b2 is dense-in-itself(b1);
:: TOPGEN_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct holds
Fr {} b1 = {};
:: TOPGEN_1:funcreg 4
registration
let a1 be TopSpace-like TopStruct;
let a2 be open closed Element of bool the carrier of a1;
cluster Fr a2 -> empty;
end;
:: TOPGEN_1:exreg 4
registration
let a1 be non empty TopSpace-like non discrete TopStruct;
cluster non open Element of bool the carrier of a1;
end;
:: TOPGEN_1:exreg 5
registration
let a1 be non empty TopSpace-like non discrete TopStruct;
cluster non closed Element of bool the carrier of a1;
end;
:: TOPGEN_1:funcreg 5
registration
let a1 be non empty TopSpace-like non discrete TopStruct;
let a2 be non open Element of bool the carrier of a1;
cluster Fr a2 -> non empty;
end;
:: TOPGEN_1:funcreg 6
registration
let a1 be non empty TopSpace-like non discrete TopStruct;
let a2 be non closed Element of bool the carrier of a1;
cluster Fr a2 -> non empty;
end;
:: TOPGEN_1:attrnot 5 => TOPGEN_1:attr 5
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is perfect means
a2 is closed(a1) & a2 is dense-in-itself(a1);
end;
:: TOPGEN_1:dfs 10
definiens
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is perfect
it is sufficient to prove
thus a2 is closed(a1) & a2 is dense-in-itself(a1);
:: TOPGEN_1:def 10
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is perfect(b1)
iff
b2 is closed(b1) & b2 is dense-in-itself(b1);
:: TOPGEN_1:condreg 2
registration
let a1 be TopSpace-like TopStruct;
cluster perfect -> closed dense-in-itself (Element of bool the carrier of a1);
end;
:: TOPGEN_1:condreg 3
registration
let a1 be TopSpace-like TopStruct;
cluster closed dense-in-itself -> perfect (Element of bool the carrier of a1);
end;
:: TOPGEN_1:th 42
theorem
for b1 being TopSpace-like TopStruct holds
Der {} b1 = {} b1;
:: TOPGEN_1:th 43
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is perfect(b1)
iff
Der b2 = b2;
:: TOPGEN_1:th 44
theorem
for b1 being TopSpace-like TopStruct holds
{} b1 is perfect(b1);
:: TOPGEN_1:condreg 4
registration
let a1 be TopSpace-like TopStruct;
cluster empty -> perfect (Element of bool the carrier of a1);
end;
:: TOPGEN_1:exreg 6
registration
let a1 be TopSpace-like TopStruct;
cluster perfect Element of bool the carrier of a1;
end;
:: TOPGEN_1:attrnot 6 => TOPGEN_1:attr 6
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is scattered means
for b1 being Element of bool the carrier of a1
st b1 is not empty & b1 c= a2
holds b1 is not dense-in-itself(a1);
end;
:: TOPGEN_1:dfs 11
definiens
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is scattered
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 is not empty & b1 c= a2
holds b1 is not dense-in-itself(a1);
:: TOPGEN_1:def 11
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is scattered(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 is not empty & b3 c= b2
holds b3 is not dense-in-itself(b1);
:: TOPGEN_1:condreg 5
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty scattered -> non dense-in-itself (Element of bool the carrier of a1);
end;
:: TOPGEN_1:condreg 6
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty dense-in-itself -> non scattered (Element of bool the carrier of a1);
end;
:: TOPGEN_1:th 45
theorem
for b1 being TopSpace-like TopStruct holds
{} b1 is scattered(b1);
:: TOPGEN_1:condreg 7
registration
let a1 be TopSpace-like TopStruct;
cluster empty -> scattered (Element of bool the carrier of a1);
end;
:: TOPGEN_1:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T1
holds ex b2, b3 being Element of bool the carrier of b1 st
b2 \/ b3 = [#] b1 & b2 misses b3 & b2 is perfect(b1) & b3 is scattered(b1);
:: TOPGEN_1:funcreg 7
registration
let a1 be TopSpace-like discrete TopStruct;
let a2 be Element of bool the carrier of a1;
cluster Fr a2 -> empty;
end;
:: TOPGEN_1:condreg 8
registration
let a1 be TopSpace-like discrete TopStruct;
cluster -> open closed (Element of bool the carrier of a1);
end;
:: TOPGEN_1:th 47
theorem
for b1 being TopSpace-like discrete TopStruct
for b2 being Element of bool the carrier of b1 holds
Der b2 = {};
:: TOPGEN_1:funcreg 8
registration
let a1 be non empty TopSpace-like discrete TopStruct;
let a2 be Element of bool the carrier of a1;
cluster Der a2 -> empty;
end;
:: TOPGEN_1:funcnot 4 => TOPGEN_1:func 4
definition
let a1 be TopSpace-like TopStruct;
func density A1 -> cardinal set means
(ex b1 being Element of bool the carrier of a1 st
b1 is dense(a1) & it = Card b1) &
(for b1 being Element of bool the carrier of a1
st b1 is dense(a1)
holds it c= Card b1);
end;
:: TOPGEN_1:def 12
theorem
for b1 being TopSpace-like TopStruct
for b2 being cardinal set holds
b2 = density b1
iff
(ex b3 being Element of bool the carrier of b1 st
b3 is dense(b1) & b2 = Card b3) &
(for b3 being Element of bool the carrier of b1
st b3 is dense(b1)
holds b2 c= Card b3);
:: TOPGEN_1:attrnot 7 => TOPGEN_1:attr 7
definition
let a1 be TopSpace-like TopStruct;
attr a1 is separable means
density a1 c= alef 0;
end;
:: TOPGEN_1:dfs 13
definiens
let a1 be TopSpace-like TopStruct;
To prove
a1 is separable
it is sufficient to prove
thus density a1 c= alef 0;
:: TOPGEN_1:def 13
theorem
for b1 being TopSpace-like TopStruct holds
b1 is separable
iff
density b1 c= alef 0;
:: TOPGEN_1:th 49
theorem
for b1 being TopSpace-like countable TopStruct holds
b1 is separable;
:: TOPGEN_1:condreg 9
registration
cluster TopSpace-like countable -> separable (TopStruct);
end;
:: TOPGEN_1:th 50
theorem
for b1 being Element of bool the carrier of R^1
st b1 = RAT
holds b1 ` = IRRAT;
:: TOPGEN_1:th 51
theorem
for b1 being Element of bool the carrier of R^1
st b1 = IRRAT
holds b1 ` = RAT;
:: TOPGEN_1:th 52
theorem
for b1 being Element of bool the carrier of R^1
st b1 = RAT
holds Int b1 = {};
:: TOPGEN_1:th 53
theorem
for b1 being Element of bool the carrier of R^1
st b1 = IRRAT
holds Int b1 = {};
:: TOPGEN_1:th 54
theorem
for b1 being Element of bool the carrier of R^1
st b1 = RAT
holds b1 is dense(R^1);
:: TOPGEN_1:th 55
theorem
for b1 being Element of bool the carrier of R^1
st b1 = IRRAT
holds b1 is dense(R^1);
:: TOPGEN_1:th 56
theorem
for b1 being Element of bool the carrier of R^1
st b1 = RAT
holds b1 is boundary(R^1);
:: TOPGEN_1:th 57
theorem
for b1 being Element of bool the carrier of R^1
st b1 = IRRAT
holds b1 is boundary(R^1);
:: TOPGEN_1:th 58
theorem
for b1 being Element of bool the carrier of R^1
st b1 = REAL
holds b1 is boundary(not R^1);
:: TOPGEN_1:th 59
theorem
ex b1, b2 being Element of bool the carrier of R^1 st
b1 is boundary(R^1) & b2 is boundary(R^1) & b1 \/ b2 is boundary(not R^1);