:: On $T_{1}$ Reflex of Topological Space
:: by Adam Naumowicz and Mariusz {\L}api\'nski
::
:: Received March 7, 1998
:: Copyright (c) 1998-2021 Association of Mizar Users

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

proof end;

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 )

proof end;

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

proof end;

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

proof end;

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

proof end;

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 )

proof end;

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 ) )

proof end;

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)}

proof end;

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} )

proof end;

:: 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)

proof end;

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

proof end;

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) );