Article URYSOHN3, MML version 4.99.1005
:: URYSOHN3:th 1
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Element of NAT holds
ex 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);
:: URYSOHN3:modenot 1 => URYSOHN3:mode 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of bool the carrier of a1;
let a4 be Element of NAT;
assume a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
mode Drizzle of A2,A3,A4 -> Function-like quasi_total Relation of dyadic a4,bool the carrier of a1 means
a2 c= it . 0 &
a3 = ([#] a1) \ (it . 1) &
(for b1, b2 being Element of dyadic a4
st b1 < b2
holds it . b1 is open(a1) & it . b2 is open(a1) & Cl (it . b1) c= it . b2);
end;
:: URYSOHN3:dfs 1
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of bool the carrier of a1;
let a4 be Element of NAT;
let a5 be Function-like quasi_total Relation of dyadic a4,bool the carrier of a1;
To prove
a5 is Drizzle of a2,a3,a4
it is sufficient to prove
thus a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
thus a2 c= a5 . 0 &
a3 = ([#] a1) \ (a5 . 1) &
(for b1, b2 being Element of dyadic a4
st b1 < b2
holds a5 . b1 is open(a1) & a5 . b2 is open(a1) & Cl (a5 . b1) c= a5 . b2);
:: URYSOHN3:def 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of NAT
st b1 is being_T4 & b2 <> {} & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
for b5 being Function-like quasi_total Relation of dyadic b4,bool the carrier of b1 holds
b5 is Drizzle of b2,b3,b4
iff
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);
:: URYSOHN3:th 3
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
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 Drizzle of b2,b3,b4 holds
ex b6 being Drizzle of b2,b3,b4 + 1 st
for b7 being Element of dyadic (b4 + 1)
st b7 in dyadic b4
holds b6 . b7 = b5 . b7;
:: URYSOHN3:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of NAT
for b5 being Drizzle of b2,b3,b4 holds
b5 is Element of PFuncs(DYADIC,bool the carrier of b1);
:: URYSOHN3:funcnot 1 => URYSOHN3:func 1
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of NAT,PFuncs(a1,a2);
let a4 be Element of NAT;
redefine func a3 . a4 -> Element of PFuncs(a1,a2);
end;
:: URYSOHN3:th 5
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
holds ex b4 being Functional_Sequence of DYADIC,bool the carrier of b1 st
for b5 being Element of NAT holds
b4 . b5 is Drizzle of b2,b3,b5 &
(for b6 being Element of proj1 (b4 . b5) holds
(b4 . b5) . b6 = (b4 . (b5 + 1)) . b6);
:: URYSOHN3:modenot 2 => URYSOHN3:mode 2
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 closed(a1) & a3 is closed(a1) & a2 misses a3;
mode Rain of A2,A3 -> Functional_Sequence of DYADIC,bool the carrier of a1 means
for b1 being Element of NAT holds
it . b1 is Drizzle of a2,a3,b1 &
(for b2 being Element of proj1 (it . b1) holds
(it . b1) . b2 = (it . (b1 + 1)) . b2);
end;
:: URYSOHN3:dfs 2
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of bool the carrier of a1;
let a4 be Functional_Sequence of DYADIC,bool the carrier of a1;
To prove
a4 is Rain of a2,a3
it is sufficient to prove
thus a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
thus for b1 being Element of NAT holds
a4 . b1 is Drizzle of a2,a3,b1 &
(for b2 being Element of proj1 (a4 . b1) holds
(a4 . b1) . b2 = (a4 . (b1 + 1)) . b2);
:: URYSOHN3:def 2
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 closed(b1) & b3 is closed(b1) & b2 misses b3
for b4 being Functional_Sequence of DYADIC,bool the carrier of b1 holds
b4 is Rain of b2,b3
iff
for b5 being Element of NAT holds
b4 . b5 is Drizzle of b2,b3,b5 &
(for b6 being Element of proj1 (b4 . b5) holds
(b4 . b5) . b6 = (b4 . (b5 + 1)) . b6);
:: URYSOHN3:funcnot 2 => URYSOHN3:func 2
definition
let a1 be Element of REAL;
assume a1 in DYADIC;
func inf_number_dyadic A1 -> Element of NAT means
(a1 in dyadic 0 implies it = 0) &
(it = 0 implies a1 in dyadic 0) &
(for b1 being Element of NAT
st a1 in dyadic (b1 + 1) & not a1 in dyadic b1
holds it = b1 + 1);
end;
:: URYSOHN3:def 3
theorem
for b1 being Element of REAL
st b1 in DYADIC
for b2 being Element of NAT holds
b2 = inf_number_dyadic b1
iff
(b1 in dyadic 0 implies b2 = 0) &
(b2 = 0 implies b1 in dyadic 0) &
(for b3 being Element of NAT
st b1 in dyadic (b3 + 1) & not b1 in dyadic b3
holds b2 = b3 + 1);
:: URYSOHN3:th 6
theorem
for b1 being Element of REAL
st b1 in DYADIC
holds b1 in dyadic inf_number_dyadic b1;
:: URYSOHN3:th 7
theorem
for b1 being Element of REAL
st b1 in DYADIC
for b2 being Element of NAT
st inf_number_dyadic b1 <= b2
holds b1 in dyadic b2;
:: URYSOHN3:th 8
theorem
for b1 being Element of REAL
for b2 being Element of NAT
st b1 in dyadic b2
holds inf_number_dyadic b1 <= b2;
:: URYSOHN3:th 9
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
st b5 in DYADIC
for b6 being Element of NAT holds
(b4 . inf_number_dyadic b5) . b5 = (b4 . ((inf_number_dyadic b5) + b6)) . b5;
:: URYSOHN3:th 10
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
st b5 in DYADIC
holds ex b6 being Element of bool the carrier of b1 st
for b7 being Element of NAT
st b5 in dyadic b7
holds b6 = (b4 . b7) . b5;
:: URYSOHN3:th 11
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3 holds
ex b5 being Function-like quasi_total Relation of DOM,bool the carrier of b1 st
for b6 being Element of REAL
st b6 in DOM
holds (b6 in halfline 0 implies b5 . b6 = {}) &
(b6 in right_open_halfline 1 implies b5 . b6 = the carrier of b1) &
(b6 in DYADIC implies for b7 being Element of NAT
st b6 in dyadic b7
holds b5 . b6 = (b4 . b7) . b6);
:: URYSOHN3:funcnot 3 => URYSOHN3:func 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of bool the carrier of a1;
let a4 be Rain of a2,a3;
assume a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
func Tempest A4 -> Function-like quasi_total Relation of DOM,bool the carrier of a1 means
for b1 being Element of REAL
st b1 in DOM
holds (b1 in halfline 0 implies it . b1 = {}) &
(b1 in right_open_halfline 1 implies it . b1 = the carrier of a1) &
(b1 in DYADIC implies for b2 being Element of NAT
st b1 in dyadic b2
holds it . b1 = (a4 . b2) . b1);
end;
:: URYSOHN3:def 4
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 closed(b1) & b3 is closed(b1) & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Function-like quasi_total Relation of DOM,bool the carrier of b1 holds
b5 = Tempest b4
iff
for b6 being Element of REAL
st b6 in DOM
holds (b6 in halfline 0 implies b5 . b6 = {}) &
(b6 in right_open_halfline 1 implies b5 . b6 = the carrier of b1) &
(b6 in DYADIC implies for b7 being Element of NAT
st b6 in dyadic b7
holds b5 . b6 = (b4 . b7) . b6);
:: URYSOHN3:funcnot 4 => URYSOHN3:func 4
definition
let a1 be non empty set;
let a2 be TopSpace-like TopStruct;
let a3 be Function-like quasi_total Relation of a1,bool the carrier of a2;
let a4 be Element of a1;
redefine func a3 . a4 -> Element of bool the carrier of a2;
end;
:: URYSOHN3:th 12
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
for b6 being Element of bool the carrier of b1
st b6 = (Tempest b4) . b5 & b5 in DOM
holds b6 is open(b1);
:: URYSOHN3:th 13
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5, b6 being Element of REAL
st b5 in DOM & b6 in DOM & b5 < b6
for b7 being Element of bool the carrier of b1
st b7 = (Tempest b4) . b5
holds Cl b7 c= (Tempest b4) . b6;
:: URYSOHN3:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1 holds
ex b6 being Element of bool ExtREAL st
for b7 being set holds
b7 in b6
iff
b7 in DYADIC &
(for b8 being Element of REAL
st b8 = b7
holds not b5 in (Tempest b4) . b8);
:: URYSOHN3:funcnot 5 => URYSOHN3:func 5
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of bool the carrier of a1;
let a4 be Rain of a2,a3;
let a5 be Element of the carrier of a1;
func Rainbow(A5,A4) -> Element of bool ExtREAL means
for b1 being set holds
b1 in it
iff
b1 in DYADIC &
(for b2 being Element of REAL
st b2 = b1
holds not a5 in (Tempest a4) . b2);
end;
:: URYSOHN3:def 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1
for b6 being Element of bool ExtREAL holds
b6 = Rainbow(b5,b4)
iff
for b7 being set holds
b7 in b6
iff
b7 in DYADIC &
(for b8 being Element of REAL
st b8 = b7
holds not b5 in (Tempest b4) . b8);
:: URYSOHN3:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1 holds
Rainbow(b5,b4) c= DYADIC;
:: URYSOHN3:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3 holds
ex b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
for b6 being Element of the carrier of b1 holds
(Rainbow(b6,b4) = {} implies b5 . b6 = 0) &
(for b7 being non empty Element of bool ExtREAL
st b7 = Rainbow(b6,b4)
holds b5 . b6 = sup b7);
:: URYSOHN3:funcnot 6 => URYSOHN3:func 6
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of bool the carrier of a1;
let a4 be Rain of a2,a3;
func Thunder A4 -> Function-like quasi_total Relation of the carrier of a1,the carrier of R^1 means
for b1 being Element of the carrier of a1 holds
(Rainbow(b1,a4) = {} implies it . b1 = 0) &
(for b2 being non empty Element of bool ExtREAL
st b2 = Rainbow(b1,a4)
holds it . b1 = sup b2);
end;
:: URYSOHN3:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 holds
b5 = Thunder b4
iff
for b6 being Element of the carrier of b1 holds
(Rainbow(b6,b4) = {} implies b5 . b6 = 0) &
(for b7 being non empty Element of bool ExtREAL
st b7 = Rainbow(b6,b4)
holds b5 . b6 = sup b7);
:: URYSOHN3:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1
for b6 being non empty Element of bool ExtREAL
st b6 = Rainbow(b5,b4)
for b7 being Element of ExtREAL
st b7 = 1
holds 0. <= sup b6 & sup b6 <= b7;
:: URYSOHN3:th 18
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of DOM
for b6 being Element of the carrier of b1
st (Thunder b4) . b6 < b5
holds b6 in (Tempest b4) . b5;
:: URYSOHN3:th 19
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
st b5 in DYADIC \/ right_open_halfline 1 & 0 < b5
for b6 being Element of the carrier of b1
st b6 in (Tempest b4) . b5
holds (Thunder b4) . b6 <= b5;
:: URYSOHN3:th 20
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of DOM
st 0 < b5
for b6 being Element of the carrier of b1
st b5 < (Thunder b4) . b6
holds not b6 in (Tempest b4) . b5;
:: URYSOHN3:th 21
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3 holds
Thunder b4 is continuous(b1, R^1) &
(for b5 being Element of the carrier of b1 holds
0 <= (Thunder b4) . b5 & (Thunder b4) . b5 <= 1 & (b5 in b2 implies (Thunder b4) . b5 = 0) & (b5 in b3 implies (Thunder b4) . b5 = 1));
:: URYSOHN3:th 22
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 <> {} & b2 misses b3
holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
b4 is continuous(b1, R^1) &
(for b5 being Element of the carrier of b1 holds
0 <= b4 . b5 & b4 . b5 <= 1 & (b5 in b2 implies b4 . b5 = 0) & (b5 in b3 implies b4 . b5 = 1));
:: URYSOHN3:th 23
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 misses b3
holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
b4 is continuous(b1, R^1) &
(for b5 being Element of the carrier of b1 holds
0 <= b4 . b5 & b4 . b5 <= 1 & (b5 in b2 implies b4 . b5 = 0) & (b5 in b3 implies b4 . b5 = 1));
:: URYSOHN3:th 24
theorem
for b1 being non empty TopSpace-like compact being_T2 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
st b2 misses b3
holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
b4 is continuous(b1, R^1) &
(for b5 being Element of the carrier of b1 holds
0 <= b4 . b5 & b4 . b5 <= 1 & (b5 in b2 implies b4 . b5 = 0) & (b5 in b3 implies b4 . b5 = 1));