:: On $T_{1}$ Reflex of Topological Space
:: by Adam Naumowicz and Mariusz {\L}api\'nski
::
:: 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 () 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 ()
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;
correctness
coherence ;
;
end;

:: deftheorem defines T_1-reflex T_1TOPSP:def 2 :
for T being non empty TopSpace holds T_1-reflex T = space ;

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;
coherence by Th4;
end;

registration
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,() equals :: T_1TOPSP:def 3
Proj ;
correctness
coherence
Proj is continuous Function of T,()
;
;
end;

:: deftheorem defines T_1-reflect T_1TOPSP:def 3 :
for T being non empty TopSpace holds T_1-reflect T = Proj ;

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,) 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 () 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 (),T1 st f = h * ()
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 (),() means :: T_1TOPSP:def 4
() * f = it * ();
existence
ex b1 being continuous Function of (),() st () * f = b1 * ()
by Th8;
uniqueness
for b1, b2 being continuous Function of (),() st () * f = b1 * () & () * f = b2 * () 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 (),() holds
( b4 = T_1-reflex f iff () * f = b4 * () );