:: Urysohn Lemma
:: by J\'ozef Bia{\l}as and Yatsuka Nakamura
::
:: Received February 16, 2001
:: Copyright (c) 2001-2011 Association of Mizar Users

begin

Lm1: for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
ex G being Function of (dyadic 0),(bool the carrier of T) st
( A c= G . 0 & B = ([#] T) \ (G . 1) & ( for r1, r2 being Element of dyadic 0 st r1 < r2 holds
( G . r1 is open & G . r2 is open & Cl (G . r1) c= G . r2 ) ) )

proof end;

theorem Th1: :: URYSOHN3:1

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for n being Element of NAT ex G being Function of (dyadic n),(bool the carrier of T) st
( A c= G . 0 & B = ([#] T) \ (G . 1) & ( for r1, r2 being Element of dyadic n st r1 < r2 holds
( G . r1 is open & G . r2 is open & Cl (G . r1) c= G . r2 ) ) )

proof end;

definition

let T be non empty TopSpace;
let A, B be Subset of T;
let n be Element of NAT ;
assume A1: ( T is normal & A <> {} & A is closed & B is closed & A misses B ) ;
mode Drizzle of A,B,n -> Function of (dyadic n),(bool the carrier of T) means :Def1: :: URYSOHN3:def 1
( A c= it . 0 & B = ([#] T) \ (it . 1) & ( for r1, r2 being Element of dyadic n st r1 < r2 holds
( it . r1 is open & it . r2 is open & Cl (it . r1) c= it . r2 ) ) );
existence
ex b₁ being Function of (dyadic n),(bool the carrier of T) st
( A c= b₁ . 0 & B = ([#] T) \ (b₁ . 1) & ( for r1, r2 being Element of dyadic n st r1 < r2 holds
( b₁ . r1 is open & b₁ . r2 is open & Cl (b₁ . r1) c= b₁ . r2 ) ) ) by A1, Th1;

end;

:: deftheorem Def1 defines Drizzle URYSOHN3:def 1 :
for T being non empty TopSpace
for A, B being Subset of T
for n being Element of NAT st T is normal & A <> {} & A is closed & B is closed & A misses B holds
for b₅ being Function of (dyadic n),(bool the carrier of T) holds
( b₅ is Drizzle of A,B,n iff ( A c= b₅ . 0 & B = ([#] T) \ (b₅ . 1) & ( for r1, r2 being Element of dyadic n st r1 < r2 holds
( b₅ . r1 is open & b₅ . r2 is open & Cl (b₅ . r1) c= b₅ . r2 ) ) ) );

theorem :: URYSOHN3:2

canceled;

theorem Th3: :: URYSOHN3:3

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for n being Element of NAT
for G being Drizzle of A,B,n ex F being Drizzle of A,B,n + 1 st
for r being Element of dyadic (n + 1) st r in dyadic n holds
F . r = G . r

proof end;

theorem Th4: :: URYSOHN3:4

for T being non empty TopSpace
for A, B being Subset of T
for n being Element of NAT
for S being Drizzle of A,B,n holds S is Element of PFuncs (DYADIC,(bool the carrier of T))

proof end;

definition

let A, B be non empty set ;
let F be Function of NAT,(PFuncs (A,B));
let n be Element of NAT ;
:: original: .
redefine func F . n -> Element of PFuncs (A,B);
coherence
F . n is Element of PFuncs (A,B) by FUNCT_2:7;

end;

theorem Th5: :: URYSOHN3:5

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
ex F being Functional_Sequence of DYADIC,(bool the carrier of T) st
for n being Element of NAT holds
( F . n is Drizzle of A,B,n & ( for r being Element of dom (F . n) holds (F . n) . r = (F . (n + 1)) . r ) )

proof end;

definition

let T be non empty TopSpace;
let A, B be Subset of T;
assume A1: ( T is normal & A <> {} & A is closed & B is closed & A misses B ) ;
mode Rain of A,B -> Functional_Sequence of DYADIC,(bool the carrier of T) means :Def2: :: URYSOHN3:def 2
for n being Element of NAT holds
( it . n is Drizzle of A,B,n & ( for r being Element of dom (it . n) holds (it . n) . r = (it . (n + 1)) . r ) );
existence
ex b₁ being Functional_Sequence of DYADIC,(bool the carrier of T) st
for n being Element of NAT holds
( b₁ . n is Drizzle of A,B,n & ( for r being Element of dom (b₁ . n) holds (b₁ . n) . r = (b₁ . (n + 1)) . r ) ) by A1, Th5;

end;

:: deftheorem Def2 defines Rain URYSOHN3:def 2 :
for T being non empty TopSpace
for A, B being Subset of T st T is normal & A <> {} & A is closed & B is closed & A misses B holds
for b₄ being Functional_Sequence of DYADIC,(bool the carrier of T) holds
( b₄ is Rain of A,B iff for n being Element of NAT holds
( b₄ . n is Drizzle of A,B,n & ( for r being Element of dom (b₄ . n) holds (b₄ . n) . r = (b₄ . (n + 1)) . r ) ) );

definition

let x be Real;
assume A1: x in DYADIC ;
func inf_number_dyadic x -> Element of NAT means :Def3: :: URYSOHN3:def 3
( ( x in dyadic 0 implies it = 0 ) & ( it = 0 implies x in dyadic 0 ) & ( for n being Element of NAT st x in dyadic (n + 1) & not x in dyadic n holds
it = n + 1 ) );
existence
ex b₁ being Element of NAT st
( ( x in dyadic 0 implies b₁ = 0 ) & ( b₁ = 0 implies x in dyadic 0 ) & ( for n being Element of NAT st x in dyadic (n + 1) & not x in dyadic n holds
b₁ = n + 1 ) )

proof end;

uniqueness
for b₁, b₂ being Element of NAT holds
not ( ( x in dyadic 0 implies b₁ = 0 ) & ( b₁ = 0 implies x in dyadic 0 ) & ( for n being Element of NAT st x in dyadic (n + 1) & not x in dyadic n holds
b₁ = n + 1 ) & ( x in dyadic 0 implies b₂ = 0 ) & ( b₂ = 0 implies x in dyadic 0 ) & ( for n being Element of NAT st x in dyadic (n + 1) & not x in dyadic n holds
b₂ = n + 1 ) & not b₁ = b₂ )

proof end;

end;

:: deftheorem Def3 defines inf_number_dyadic URYSOHN3:def 3 :
for x being Real st x in DYADIC holds
for b₂ being Element of NAT holds
( b₂ = inf_number_dyadic x iff ( ( x in dyadic 0 implies b₂ = 0 ) & ( b₂ = 0 implies x in dyadic 0 ) & ( for n being Element of NAT st x in dyadic (n + 1) & not x in dyadic n holds
b₂ = n + 1 ) ) );

theorem Th6: :: URYSOHN3:6

for x being Real st x in DYADIC holds
x in dyadic (inf_number_dyadic x)

proof end;

theorem Th7: :: URYSOHN3:7

for x being Real st x in DYADIC holds
for n being Element of NAT st inf_number_dyadic x <= n holds
x in dyadic n

proof end;

theorem Th8: :: URYSOHN3:8

for x being Real
for n being Element of NAT st x in dyadic n holds
inf_number_dyadic x <= n

proof end;

theorem Th9: :: URYSOHN3:9

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B
for x being Real st x in DYADIC holds
for n being Element of NAT holds (G . (inf_number_dyadic x)) . x = (G . ((inf_number_dyadic x) + n)) . x

proof end;

theorem Th10: :: URYSOHN3:10

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B
for x being Real st x in DYADIC holds
ex y being Subset of T st
for n being Element of NAT st x in dyadic n holds
y = (G . n) . x

proof end;

theorem Th11: :: URYSOHN3:11

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B ex F being Function of DOM,(bool the carrier of T) st
for x being Real st x in DOM holds
( ( x in halfline 0 implies F . x = {} ) & ( x in right_open_halfline 1 implies F . x = the carrier of T ) & ( x in DYADIC implies for n being Element of NAT st x in dyadic n holds
F . x = (G . n) . x ) )

proof end;

definition

let T be non empty TopSpace;
let A, B be Subset of T;
assume A1: ( T is normal & A <> {} & A is closed & B is closed & A misses B ) ;
let R be Rain of A,B;
func Tempest R -> Function of DOM,(bool the carrier of T) means :Def4: :: URYSOHN3:def 4
for x being Real st x in DOM holds
( ( x in halfline 0 implies it . x = {} ) & ( x in right_open_halfline 1 implies it . x = the carrier of T ) & ( x in DYADIC implies for n being Element of NAT st x in dyadic n holds
it . x = (R . n) . x ) );
existence
ex b₁ being Function of DOM,(bool the carrier of T) st
for x being Real st x in DOM holds
( ( x in halfline 0 implies b₁ . x = {} ) & ( x in right_open_halfline 1 implies b₁ . x = the carrier of T ) & ( x in DYADIC implies for n being Element of NAT st x in dyadic n holds
b₁ . x = (R . n) . x ) ) by A1, Th11;
uniqueness
for b₁, b₂ being Function of DOM,(bool the carrier of T) st ( for x being Real st x in DOM holds
( ( x in halfline 0 implies b₁ . x = {} ) & ( x in right_open_halfline 1 implies b₁ . x = the carrier of T ) & ( x in DYADIC implies for n being Element of NAT st x in dyadic n holds
b₁ . x = (R . n) . x ) ) ) & ( for x being Real st x in DOM holds
( ( x in halfline 0 implies b₂ . x = {} ) & ( x in right_open_halfline 1 implies b₂ . x = the carrier of T ) & ( x in DYADIC implies for n being Element of NAT st x in dyadic n holds
b₂ . x = (R . n) . x ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines Tempest URYSOHN3:def 4 :
for T being non empty TopSpace
for A, B being Subset of T st T is normal & A <> {} & A is closed & B is closed & A misses B holds
for R being Rain of A,B
for b₅ being Function of DOM,(bool the carrier of T) holds
( b₅ = Tempest R iff for x being Real st x in DOM holds
( ( x in halfline 0 implies b₅ . x = {} ) & ( x in right_open_halfline 1 implies b₅ . x = the carrier of T ) & ( x in DYADIC implies for n being Element of NAT st x in dyadic n holds
b₅ . x = (R . n) . x ) ) );

definition

let X be non empty set ;
let T be TopSpace;
let F be Function of X,(bool the carrier of T);
let x be Element of X;
:: original: .
redefine func F . x -> Subset of T;
correctness
coherence
F . x is Subset of T;

proof end;

end;

theorem Th12: :: URYSOHN3:12

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B
for r being Real
for C being Subset of T st C = (Tempest G) . r & r in DOM holds
C is open

proof end;

theorem Th13: :: URYSOHN3:13

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B
for r1, r2 being Real st r1 in DOM & r2 in DOM & r1 < r2 holds
for C being Subset of T st C = (Tempest G) . r1 holds
Cl C c= (Tempest G) . r2

proof end;

theorem Th14: :: URYSOHN3:14

for T being non empty TopSpace
for A, B being Subset of T
for G being Rain of A,B
for p being Point of T ex R being Subset of ExtREAL st
for x being set holds
( x in R iff ( x in DYADIC & ( for s being Real st s = x holds
not p in (Tempest G) . s ) ) )

proof end;

definition

let T be non empty TopSpace;
let A, B be Subset of T;
let R be Rain of A,B;
let p be Point of T;
func Rainbow (p,R) -> Subset of ExtREAL means :Def5: :: URYSOHN3:def 5
for x being set holds
( x in it iff ( x in DYADIC & ( for s being Real st s = x holds
not p in (Tempest R) . s ) ) );
existence
ex b₁ being Subset of ExtREAL st
for x being set holds
( x in b₁ iff ( x in DYADIC & ( for s being Real st s = x holds
not p in (Tempest R) . s ) ) ) by Th14;
uniqueness
for b₁, b₂ being Subset of ExtREAL st ( for x being set holds
( x in b₁ iff ( x in DYADIC & ( for s being Real st s = x holds
not p in (Tempest R) . s ) ) ) ) & ( for x being set holds
( x in b₂ iff ( x in DYADIC & ( for s being Real st s = x holds
not p in (Tempest R) . s ) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines Rainbow URYSOHN3:def 5 :
for T being non empty TopSpace
for A, B being Subset of T
for R being Rain of A,B
for p being Point of T
for b₆ being Subset of ExtREAL holds
( b₆ = Rainbow (p,R) iff for x being set holds
( x in b₆ iff ( x in DYADIC & ( for s being Real st s = x holds
not p in (Tempest R) . s ) ) ) );

theorem Th15: :: URYSOHN3:15

for T being non empty TopSpace
for A, B being Subset of T
for G being Rain of A,B
for p being Point of T holds Rainbow (p,G) c= DYADIC

proof end;

theorem Th16: :: URYSOHN3:16

for T being non empty TopSpace
for A, B being Subset of T
for R being Rain of A,B ex F being Function of T,R^1 st
for p being Point of T holds
( ( Rainbow (p,R) = {} implies F . p = 0 ) & ( for S being non empty Subset of ExtREAL st S = Rainbow (p,R) holds
F . p = sup S ) )

proof end;

definition

let T be non empty TopSpace;
let A, B be Subset of T;
let R be Rain of A,B;
func Thunder R -> Function of T,R^1 means :Def6: :: URYSOHN3:def 6
for p being Point of T holds
( ( Rainbow (p,R) = {} implies it . p = 0 ) & ( for S being non empty Subset of ExtREAL st S = Rainbow (p,R) holds
it . p = sup S ) );
existence
ex b₁ being Function of T,R^1 st
for p being Point of T holds
( ( Rainbow (p,R) = {} implies b₁ . p = 0 ) & ( for S being non empty Subset of ExtREAL st S = Rainbow (p,R) holds
b₁ . p = sup S ) ) by Th16;
uniqueness
for b₁, b₂ being Function of T,R^1 st ( for p being Point of T holds
( ( Rainbow (p,R) = {} implies b₁ . p = 0 ) & ( for S being non empty Subset of ExtREAL st S = Rainbow (p,R) holds
b₁ . p = sup S ) ) ) & ( for p being Point of T holds
( ( Rainbow (p,R) = {} implies b₂ . p = 0 ) & ( for S being non empty Subset of ExtREAL st S = Rainbow (p,R) holds
b₂ . p = sup S ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines Thunder URYSOHN3:def 6 :
for T being non empty TopSpace
for A, B being Subset of T
for R being Rain of A,B
for b₅ being Function of T,R^1 holds
( b₅ = Thunder R iff for p being Point of T holds
( ( Rainbow (p,R) = {} implies b₅ . p = 0 ) & ( for S being non empty Subset of ExtREAL st S = Rainbow (p,R) holds
b₅ . p = sup S ) ) );

theorem Th17: :: URYSOHN3:17

for T being non empty TopSpace
for A, B being Subset of T
for G being Rain of A,B
for p being Point of T
for S being non empty Subset of ExtREAL st S = Rainbow (p,G) holds
for e1 being R_eal st e1 = 1 holds
( 0. <= sup S & sup S <= e1 )

proof end;

theorem Th18: :: URYSOHN3:18

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B
for r being Element of DOM
for p being Point of T st (Thunder G) . p < r holds
p in (Tempest G) . r

proof end;

theorem Th19: :: URYSOHN3:19

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B
for r being Real st r in DYADIC \/ (right_open_halfline 1) & 0 < r holds
for p being Point of T st p in (Tempest G) . r holds
(Thunder G) . p <= r

proof end;

theorem Th20: :: URYSOHN3:20

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B
for r1 being Element of DOM st 0 < r1 holds
for p being Point of T st r1 < (Thunder G) . p holds
not p in (Tempest G) . r1

proof end;

theorem Th21: :: URYSOHN3:21

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
for G being Rain of A,B holds
( Thunder G is continuous & ( for x being Point of T holds
( 0 <= (Thunder G) . x & (Thunder G) . x <= 1 & ( x in A implies (Thunder G) . x = 0 ) & ( x in B implies (Thunder G) . x = 1 ) ) ) )

proof end;

theorem Th22: :: URYSOHN3:22

for T being non empty normal TopSpace
for A, B being closed Subset of T st A <> {} & A misses B holds
ex F being Function of T,R^1 st
( F is continuous & ( for x being Point of T holds
( 0 <= F . x & F . x <= 1 & ( x in A implies F . x = 0 ) & ( x in B implies F . x = 1 ) ) ) )

proof end;

theorem Th23: :: URYSOHN3:23

for T being non empty normal TopSpace
for A, B being closed Subset of T st A misses B holds
ex F being Function of T,R^1 st
( F is continuous & ( for x being Point of T holds
( 0 <= F . x & F . x <= 1 & ( x in A implies F . x = 0 ) & ( x in B implies F . x = 1 ) ) ) )

proof end;

theorem :: URYSOHN3:24

for T being non empty T_2 compact TopSpace
for A, B being closed Subset of T st A misses B holds
ex F being Function of T,R^1 st
( F is continuous & ( for x being Point of T holds
( 0 <= F . x & F . x <= 1 & ( x in A implies F . x = 0 ) & ( x in B implies F . x = 1 ) ) ) ) by Th23;