Article URYSOHN1, MML version 4.99.1005
:: URYSOHN1:th 1
theorem
for b1 being Element of REAL holds
b1 in halfline 0
iff
b1 < 0;
:: URYSOHN1:th 2
theorem
for b1 being Element of REAL holds
b1 in right_open_halfline 1
iff
1 < b1;
:: URYSOHN1:funcnot 1 => URYSOHN1:func 1
definition
let a1 be natural set;
func dyadic A1 -> Element of bool REAL means
for b1 being Element of REAL holds
b1 in it
iff
ex b2 being Element of NAT st
0 <= b2 & b2 <= 2 |^ a1 & b1 = b2 / (2 |^ a1);
end;
:: URYSOHN1:def 3
theorem
for b1 being natural set
for b2 being Element of bool REAL holds
b2 = dyadic b1
iff
for b3 being Element of REAL holds
b3 in b2
iff
ex b4 being Element of NAT st
0 <= b4 & b4 <= 2 |^ b1 & b3 = b4 / (2 |^ b1);
:: URYSOHN1:funcnot 2 => URYSOHN1:func 2
definition
func DYADIC -> Element of bool REAL means
for b1 being Element of REAL holds
b1 in it
iff
ex b2 being Element of NAT st
b1 in dyadic b2;
end;
:: URYSOHN1:def 4
theorem
for b1 being Element of bool REAL holds
b1 = DYADIC
iff
for b2 being Element of REAL holds
b2 in b1
iff
ex b3 being Element of NAT st
b2 in dyadic b3;
:: URYSOHN1:funcnot 3 => URYSOHN1:func 3
definition
func DOM -> Element of bool REAL equals
((halfline 0) \/ DYADIC) \/ right_open_halfline 1;
end;
:: URYSOHN1:def 5
theorem
DOM = ((halfline 0) \/ DYADIC) \/ right_open_halfline 1;
:: URYSOHN1:funcnot 4 => URYSOHN1:func 4
definition
let a1 be TopSpace-like TopStruct;
let a2 be non empty Element of bool REAL;
let a3 be Function-like quasi_total Relation of a2,bool the carrier of a1;
let a4 be Element of a2;
redefine func a3 . a4 -> Element of bool the carrier of a1;
end;
:: URYSOHN1:th 5
theorem
for b1 being Element of NAT
for b2 being Element of REAL
st b2 in dyadic b1
holds 0 <= b2 & b2 <= 1;
:: URYSOHN1:th 6
theorem
dyadic 0 = {0,1};
:: URYSOHN1:th 7
theorem
dyadic 1 = {0,1 / 2,1};
:: URYSOHN1:funcreg 1
registration
let a1 be Element of NAT;
cluster dyadic a1 -> non empty;
end;
:: URYSOHN1:th 9
theorem
for b1 being Element of NAT holds
ex b2 being Relation-like Function-like FinSequence-like set st
dom b2 = Seg ((2 |^ b1) + 1) &
(for b3 being Element of NAT
st b3 in dom b2
holds b2 . b3 = (b3 - 1) / (2 |^ b1));
:: URYSOHN1:funcnot 5 => URYSOHN1:func 5
definition
let a1 be Element of NAT;
func dyad A1 -> Relation-like Function-like FinSequence-like set means
dom it = Seg ((2 |^ a1) + 1) &
(for b1 being Element of NAT
st b1 in dom it
holds it . b1 = (b1 - 1) / (2 |^ a1));
end;
:: URYSOHN1:def 6
theorem
for b1 being Element of NAT
for b2 being Relation-like Function-like FinSequence-like set holds
b2 = dyad b1
iff
dom b2 = Seg ((2 |^ b1) + 1) &
(for b3 being Element of NAT
st b3 in dom b2
holds b2 . b3 = (b3 - 1) / (2 |^ b1));
:: URYSOHN1:th 10
theorem
for b1 being Element of NAT holds
dom dyad b1 = Seg ((2 |^ b1) + 1) &
proj2 dyad b1 = dyadic b1;
:: URYSOHN1:funcreg 2
registration
cluster DYADIC -> non empty;
end;
:: URYSOHN1:funcreg 3
registration
cluster DOM -> non empty;
end;
:: URYSOHN1:th 11
theorem
for b1 being Element of NAT holds
dyadic b1 c= dyadic (b1 + 1);
:: URYSOHN1:th 12
theorem
for b1 being Element of NAT holds
0 in dyadic b1 & 1 in dyadic b1;
:: URYSOHN1:th 13
theorem
for b1, b2 being Element of NAT
st 0 < b2 & b2 <= 2 |^ b1
holds ((b2 * 2) - 1) / (2 |^ (b1 + 1)) in (dyadic (b1 + 1)) \ dyadic b1;
:: URYSOHN1:th 14
theorem
for b1, b2 being Element of NAT
st 0 <= b2 & b2 < 2 |^ b1
holds ((b2 * 2) + 1) / (2 |^ (b1 + 1)) in (dyadic (b1 + 1)) \ dyadic b1;
:: URYSOHN1:th 15
theorem
for b1 being Element of NAT holds
1 / (2 |^ (b1 + 1)) in (dyadic (b1 + 1)) \ dyadic b1;
:: URYSOHN1:funcnot 6 => URYSOHN1:func 6
definition
let a1 be Element of NAT;
let a2 be Element of dyadic a1;
func axis(A2,A1) -> Element of NAT means
a2 = it / (2 |^ a1);
end;
:: URYSOHN1:def 7
theorem
for b1 being Element of NAT
for b2 being Element of dyadic b1
for b3 being Element of NAT holds
b3 = axis(b2,b1)
iff
b2 = b3 / (2 |^ b1);
:: URYSOHN1:th 16
theorem
for b1 being Element of NAT
for b2 being Element of dyadic b1 holds
b2 = (axis(b2,b1)) / (2 |^ b1) & axis(b2,b1) <= 2 |^ b1;
:: URYSOHN1:th 17
theorem
for b1 being Element of NAT
for b2 being Element of dyadic b1 holds
((axis(b2,b1)) - 1) / (2 |^ b1) < b2 &
b2 < ((axis(b2,b1)) + 1) / (2 |^ b1);
:: URYSOHN1:th 18
theorem
for b1 being Element of NAT
for b2 being Element of dyadic b1 holds
((axis(b2,b1)) - 1) / (2 |^ b1) < ((axis(b2,b1)) + 1) / (2 |^ b1);
:: URYSOHN1:th 20
theorem
for b1 being Element of NAT
for b2 being Element of dyadic (b1 + 1)
st not b2 in dyadic b1
holds ((axis(b2,b1 + 1)) - 1) / (2 |^ (b1 + 1)) in dyadic b1 &
((axis(b2,b1 + 1)) + 1) / (2 |^ (b1 + 1)) in dyadic b1;
:: URYSOHN1:th 21
theorem
for b1 being Element of NAT
for b2, b3 being Element of dyadic b1
st b2 < b3
holds axis(b2,b1) < axis(b3,b1);
:: URYSOHN1:th 22
theorem
for b1 being Element of NAT
for b2, b3 being Element of dyadic b1
st b2 < b3
holds b2 <= ((axis(b3,b1)) - 1) / (2 |^ b1) &
((axis(b2,b1)) + 1) / (2 |^ b1) <= b3;
:: URYSOHN1:th 23
theorem
for b1 being Element of NAT
for b2, b3 being Element of dyadic (b1 + 1)
st b2 < b3 & not b2 in dyadic b1 & not b3 in dyadic b1
holds ((axis(b2,b1 + 1)) + 1) / (2 |^ (b1 + 1)) <= ((axis(b3,b1 + 1)) - 1) / (2 |^ (b1 + 1));
:: URYSOHN1:modenot 1 => CONNSP_2:mode 1
notation
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
synonym Nbhd of a2,a1 for a_neighborhood of a2;
end;
:: URYSOHN1:modenot 2 => CONNSP_2:mode 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
mode Nbhd of A2,A1 -> Element of bool the carrier of a1 means
ex b1 being Element of bool the carrier of a1 st
b1 is open(a1) & a2 in b1 & b1 c= it;
end;
:: URYSOHN1:dfs 6
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the carrier of a1;
To prove
a3 is a_neighborhood of a2
it is sufficient to prove
thus ex b1 being Element of bool the carrier of a1 st
b1 is open(a1) & a2 in b1 & b1 c= a3;
:: URYSOHN1:def 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 is a_neighborhood of b2
iff
ex b4 being Element of bool the carrier of b1 st
b4 is open(b1) & b2 in b4 & b4 c= b3;
:: URYSOHN1:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds ex b4 being a_neighborhood of b3 st
b4 c= b2;
:: URYSOHN1:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st for b3 being Element of the carrier of b1
st b3 in b2
holds b2 is a_neighborhood of b3
holds b2 is open(b1);
:: URYSOHN1:attrnot 1 => URYSOHN1:attr 1
definition
let a1 be TopStruct;
attr a1 is being_T1 means
for b1, b2 being Element of the carrier of a1
st b1 <> b2
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 in b3 & not b2 in b3 & b2 in b4 & not b1 in b4;
end;
:: URYSOHN1:dfs 7
definiens
let a1 be TopStruct;
To prove
a1 is being_T1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 <> b2
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 in b3 & not b2 in b3 & b2 in b4 & not b1 in b4;
:: URYSOHN1:def 9
theorem
for b1 being TopStruct holds
b1 is being_T1
iff
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ex b4, b5 being Element of bool the carrier of b1 st
b4 is open(b1) & b5 is open(b1) & b2 in b4 & not b3 in b4 & b3 in b5 & not b2 in b5;
:: URYSOHN1:prednot 1 => URYSOHN1:attr 1
notation
let a1 be TopStruct;
synonym a1 is_T1 for being_T1;
end;
:: URYSOHN1:th 27
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is being_T1
iff
for b2 being Element of the carrier of b1 holds
{b2} is closed(b1);
:: URYSOHN1:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T4
for b2, b3 being open Element of bool the carrier of b1
st b2 <> {} & Cl b2 c= b3
holds ex b4 being Element of bool the carrier of b1 st
b4 <> {} & b4 is open(b1) & Cl b2 c= b4 & Cl b4 c= b3;
:: URYSOHN1:th 29
theorem
for b1 being TopSpace-like TopStruct holds
b1 is being_T3
iff
for b2 being open Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in b2
holds ex b4 being open Element of bool the carrier of b1 st
b3 in b4 & Cl b4 c= b2;
:: URYSOHN1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T4 & b1 is being_T1
for b2 being open Element of bool the carrier of b1
st b2 <> {}
holds ex b3 being Element of bool the carrier of b1 st
b3 <> {} & Cl b3 c= b2;
:: URYSOHN1:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T4
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1) & b3 is closed(b1) & b3 <> {} & b3 c= b2
holds ex b4 being Element of bool the carrier of b1 st
b4 is open(b1) & b3 c= b4 & Cl b4 c= b2;
:: URYSOHN1:modenot 3 => URYSOHN1:mode 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of bool the carrier of a1;
assume a1 is being_T4 & a2 <> {} & a2 is open(a1) & a3 is open(a1) & Cl a2 c= a3;
mode Between of A2,A3 -> Element of bool the carrier of a1 means
it <> {} & it is open(a1) & Cl a2 c= it & Cl it c= a3;
end;
:: URYSOHN1:dfs 8
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2, a3, a4 be Element of bool the carrier of a1;
To prove
a4 is Between of a2,a3
it is sufficient to prove
thus a1 is being_T4 & a2 <> {} & a2 is open(a1) & a3 is open(a1) & Cl a2 c= a3;
thus a4 <> {} & a4 is open(a1) & Cl a2 c= a4 & Cl a4 c= a3;
:: URYSOHN1:def 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b1 is being_T4 & b2 <> {} & b2 is open(b1) & b3 is open(b1) & Cl b2 c= b3
for b4 being Element of bool the carrier of b1 holds
b4 is Between of b2,b3
iff
b4 <> {} & b4 is open(b1) & Cl b2 c= b4 & Cl b4 c= b3;
:: URYSOHN1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T4
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Element of NAT
for b5 being Function-like quasi_total Relation of dyadic b4,bool the carrier of b1
st b2 c= b5 . 0 &
b3 = ([#] b1) \ (b5 . 1) &
(for b6, b7 being Element of dyadic b4
st b6 < b7
holds b5 . b6 is open(b1) & b5 . b7 is open(b1) & Cl (b5 . b6) c= b5 . b7)
holds ex b6 being Function-like quasi_total Relation of dyadic (b4 + 1),bool the carrier of b1 st
b2 c= b6 . 0 &
b3 = ([#] b1) \ (b6 . 1) &
(for b7, b8, b9 being Element of dyadic (b4 + 1)
st b7 < b8
holds b6 . b7 is open(b1) &
b6 . b8 is open(b1) &
Cl (b6 . b7) c= b6 . b8 &
(b9 in dyadic b4 implies b6 . b9 = b5 . b9));