:: On $T_{1}$ Reflex of Topological Space
:: by Adam Naumowicz and Mariusz {\L}api\'nski
::
:: Received March 7, 1998
:: Copyright (c) 1998-2012 Association of Mizar Users


begin

theorem Th1: :: T_1TOPSP:1
for T being non empty TopSpace
for A being non empty a_partition of the carrier of T
for y being Subset of (space A) holds (Proj A) " y = union y
proofend;

theorem Th2: :: T_1TOPSP:2
for T being non empty TopSpace
for S being non empty a_partition of the carrier of T
for A being Subset of (space S)
for B being Subset of T st B = union A holds
( A is closed iff B is closed )
proofend;

theorem Th3: :: T_1TOPSP:3
for T being non empty TopSpace holds  {  A where A is a_partition of the carrier of T : A is closed  }  is Part-Family of the carrier of T
proofend;

definition
let T be non empty TopSpace;
func Closed_Partitions T ->  non empty Part-Family of the carrier of T equals :: T_1TOPSP:def 1
 {  A where A is a_partition of the carrier of T : A is closed  }  ;
coherence
  {  A where A is a_partition of the carrier of T : A is closed  }  is non empty Part-Family of the carrier of T
proofend;
end;

:: deftheorem  defines Closed_Partitions T_1TOPSP:def_1_:_
 for T being non empty TopSpace holds Closed_Partitions T =  {  A where A is a_partition of the carrier of T : A is closed  }  ;

:: T_1 reflex of a topological space
definition
let T be non empty TopSpace;
func T_1-reflex T ->  TopSpace equals :: T_1TOPSP:def 2
 space (Intersection (Closed_Partitions T));
correctness
coherence
space (Intersection (Closed_Partitions T)) is TopSpace;
;
end;

:: deftheorem  defines T_1-reflex T_1TOPSP:def_2_:_
 for T being non empty TopSpace holds T_1-reflex T = space (Intersection (Closed_Partitions T));

registration
let T be non empty TopSpace;
cluster T_1-reflex T ->  non empty strict ;
coherence
 ( T_1-reflex T is strict & not T_1-reflex T is empty ) ;
end;

theorem Th4: :: T_1TOPSP:4
for T being non empty TopSpace holds T_1-reflex T is T_1
proofend;

registration
let T be non empty TopSpace;
cluster T_1-reflex T ->  T_1 ;
coherence
 T_1-reflex T is T_1 by Th4;
end;

registration
cluster non empty TopSpace-like T_1 for TopStruct ;
existence
 ex b1 being TopSpace st
( b1 is T_1 & not b1 is empty )
proofend;
end;

definition
let T be non empty TopSpace;
func T_1-reflect T ->  continuous Function of T,(T_1-reflex T) equals :: T_1TOPSP:def 3
 Proj (Intersection (Closed_Partitions T));
correctness
coherence
Proj (Intersection (Closed_Partitions T)) is continuous Function of T,(T_1-reflex T);
;
end;

:: deftheorem  defines T_1-reflect T_1TOPSP:def_3_:_
 for T being non empty TopSpace holds T_1-reflect T = Proj (Intersection (Closed_Partitions T));

theorem Th5: :: T_1TOPSP:5
for T, T1 being non empty TopSpace
for f being continuous Function of T,T1 st T1 is T_1 holds
(  {  (f " {z}) where z is Element of T1 : z in rng f  }  is a_partition of the carrier of T & ( for A being Subset of T st A in  {  (f " {z}) where z is Element of T1 : z in rng f  }  holds
A is closed ) )
proofend;

theorem Th6: :: T_1TOPSP:6
for T, T1 being non empty TopSpace
for f being continuous Function of T,T1 st T1 is T_1 holds
for w being set
for x being Element of T st w = EqClass (x,(Intersection (Closed_Partitions T))) holds
w c= f " {(f . x)}
proofend;

theorem Th7: :: T_1TOPSP:7
for T, T1 being non empty TopSpace
for f being continuous Function of T,T1 st T1 is T_1 holds
for w being set st w in the carrier of (T_1-reflex T) holds
ex z being Element of T1 st
( z in rng f & w c= f " {z} )
proofend;

:: The theorem on factorization
theorem Th8: :: T_1TOPSP:8
for T, T1 being non empty TopSpace
for f being continuous Function of T,T1 st T1 is T_1 holds
ex h being continuous Function of (T_1-reflex T),T1 st f = h * (T_1-reflect T)
proofend;

definition
let T, S be non empty TopSpace;
let f be continuous Function of T,S;
func T_1-reflex f ->  continuous Function of (T_1-reflex T),(T_1-reflex S) means :: T_1TOPSP:def 4
(T_1-reflect S) * f = it * (T_1-reflect T);
existence
 ex b1 being continuous Function of (T_1-reflex T),(T_1-reflex S) st (T_1-reflect S) * f = b1 * (T_1-reflect T) by Th8;
uniqueness
 for b1, b2 being continuous Function of (T_1-reflex T),(T_1-reflex S) st (T_1-reflect S) * f = b1 * (T_1-reflect T) & (T_1-reflect S) * f = b2 * (T_1-reflect T) holds
b1 = b2
proofend;
end;

:: deftheorem  defines T_1-reflex T_1TOPSP:def_4_:_
 for T, S being non empty TopSpace
for f being continuous Function of T,S
for b4 being continuous Function of (T_1-reflex T),(T_1-reflex S) holds
( b4 = T_1-reflex f iff (T_1-reflect S) * f = b4 * (T_1-reflect T) );