:: On the Decomposition of the Continuity
:: by Marian Przemski
::
:: Received December 12, 1994
:: Copyright (c) 1994-2012 Association of Mizar Users


begin

definition
let T be TopStruct ;
mode alpha-set of T ->  Subset of T means :Def1: :: DECOMP_1:def 1
it c= Int (Cl (Int it));
existence
 ex b1 being Subset of T st b1 c= Int (Cl (Int b1))
proofend;
let IT be Subset of T;
attrIT is semi-open  means :Def2: :: DECOMP_1:def 2
IT c= Cl (Int IT);
attrIT is pre-open  means :Def3: :: DECOMP_1:def 3
IT c= Int (Cl IT);
attrIT is pre-semi-open  means :Def4: :: DECOMP_1:def 4
IT c= Cl (Int (Cl IT));
attrIT is semi-pre-open  means :Def5: :: DECOMP_1:def 5
IT c= (Cl (Int IT)) \/ (Int (Cl IT));
end;

:: deftheorem Def1 defines alpha-set DECOMP_1:def_1_:_
 for T being TopStruct
for b2 being Subset of T holds
( b2 is alpha-set of T iff b2 c= Int (Cl (Int b2)) );

:: deftheorem Def2 defines semi-open DECOMP_1:def_2_:_
 for T being TopStruct
for IT being Subset of T holds
( IT is semi-open iff IT c= Cl (Int IT) );

:: deftheorem Def3 defines pre-open DECOMP_1:def_3_:_
 for T being TopStruct
for IT being Subset of T holds
( IT is pre-open iff IT c= Int (Cl IT) );

:: deftheorem Def4 defines pre-semi-open DECOMP_1:def_4_:_
 for T being TopStruct
for IT being Subset of T holds
( IT is pre-semi-open iff IT c= Cl (Int (Cl IT)) );

:: deftheorem Def5 defines semi-pre-open DECOMP_1:def_5_:_
 for T being TopStruct
for IT being Subset of T holds
( IT is semi-pre-open iff IT c= (Cl (Int IT)) \/ (Int (Cl IT)) );

definition
let T be TopStruct ;
let B be Subset of T;
func sInt B ->  Subset of T equals :: DECOMP_1:def 6
B /\ (Cl (Int B));
coherence
 B /\ (Cl (Int B)) is Subset of T ;
func pInt B ->  Subset of T equals :: DECOMP_1:def 7
B /\ (Int (Cl B));
coherence
 B /\ (Int (Cl B)) is Subset of T ;
func alphaInt B ->  Subset of T equals :: DECOMP_1:def 8
B /\ (Int (Cl (Int B)));
coherence
 B /\ (Int (Cl (Int B))) is Subset of T ;
func psInt B ->  Subset of T equals :: DECOMP_1:def 9
B /\ (Cl (Int (Cl B)));
coherence
 B /\ (Cl (Int (Cl B))) is Subset of T ;
end;

:: deftheorem  defines sInt DECOMP_1:def_6_:_
 for T being TopStruct
for B being Subset of T holds sInt B = B /\ (Cl (Int B));

:: deftheorem  defines pInt DECOMP_1:def_7_:_
 for T being TopStruct
for B being Subset of T holds pInt B = B /\ (Int (Cl B));

:: deftheorem  defines alphaInt DECOMP_1:def_8_:_
 for T being TopStruct
for B being Subset of T holds alphaInt B = B /\ (Int (Cl (Int B)));

:: deftheorem  defines psInt DECOMP_1:def_9_:_
 for T being TopStruct
for B being Subset of T holds psInt B = B /\ (Cl (Int (Cl B)));

definition
let T be TopStruct ;
let B be Subset of T;
func spInt B ->  Subset of T equals :: DECOMP_1:def 10
(sInt B) \/ (pInt B);
coherence
 (sInt B) \/ (pInt B) is Subset of T ;
end;

:: deftheorem  defines spInt DECOMP_1:def_10_:_
 for T being TopStruct
for B being Subset of T holds spInt B = (sInt B) \/ (pInt B);

definition
let T be TopSpace;
funcT ^alpha  ->  Subset-Family of T equals :: DECOMP_1:def 11
 {  B where B is Subset of T : B is alpha-set of T  }  ;
coherence
  {  B where B is Subset of T : B is alpha-set of T  }  is Subset-Family of T
proofend;
func SO T ->  Subset-Family of T equals :: DECOMP_1:def 12
 {  B where B is Subset of T : B is semi-open  }  ;
coherence
  {  B where B is Subset of T : B is semi-open  }  is Subset-Family of T
proofend;
func PO T ->  Subset-Family of T equals :: DECOMP_1:def 13
 {  B where B is Subset of T : B is pre-open  }  ;
coherence
  {  B where B is Subset of T : B is pre-open  }  is Subset-Family of T
proofend;
func SPO T ->  Subset-Family of T equals :: DECOMP_1:def 14
 {  B where B is Subset of T : B is semi-pre-open  }  ;
coherence
  {  B where B is Subset of T : B is semi-pre-open  }  is Subset-Family of T
proofend;
func PSO T ->  Subset-Family of T equals :: DECOMP_1:def 15
 {  B where B is Subset of T : B is pre-semi-open  }  ;
coherence
  {  B where B is Subset of T : B is pre-semi-open  }  is Subset-Family of T
proofend;
func D(c,alpha) T ->  Subset-Family of T equals :: DECOMP_1:def 16
 {  B where B is Subset of T : Int B = alphaInt B  }  ;
coherence
  {  B where B is Subset of T : Int B = alphaInt B  }  is Subset-Family of T
proofend;
func D(c,p) T ->  Subset-Family of T equals :: DECOMP_1:def 17
 {  B where B is Subset of T : Int B = pInt B  }  ;
coherence
  {  B where B is Subset of T : Int B = pInt B  }  is Subset-Family of T
proofend;
func D(c,s) T ->  Subset-Family of T equals :: DECOMP_1:def 18
 {  B where B is Subset of T : Int B = sInt B  }  ;
coherence
  {  B where B is Subset of T : Int B = sInt B  }  is Subset-Family of T
proofend;
func D(c,ps) T ->  Subset-Family of T equals :: DECOMP_1:def 19
 {  B where B is Subset of T : Int B = psInt B  }  ;
coherence
  {  B where B is Subset of T : Int B = psInt B  }  is Subset-Family of T
proofend;
func D(alpha,p) T ->  Subset-Family of T equals :: DECOMP_1:def 20
 {  B where B is Subset of T : alphaInt B = pInt B  }  ;
coherence
  {  B where B is Subset of T : alphaInt B = pInt B  }  is Subset-Family of T
proofend;
func D(alpha,s) T ->  Subset-Family of T equals :: DECOMP_1:def 21
 {  B where B is Subset of T : alphaInt B = sInt B  }  ;
coherence
  {  B where B is Subset of T : alphaInt B = sInt B  }  is Subset-Family of T
proofend;
func D(alpha,ps) T ->  Subset-Family of T equals :: DECOMP_1:def 22
 {  B where B is Subset of T : alphaInt B = psInt B  }  ;
coherence
  {  B where B is Subset of T : alphaInt B = psInt B  }  is Subset-Family of T
proofend;
func D(p,sp) T ->  Subset-Family of T equals :: DECOMP_1:def 23
 {  B where B is Subset of T : pInt B = spInt B  }  ;
coherence
  {  B where B is Subset of T : pInt B = spInt B  }  is Subset-Family of T
proofend;
func D(p,ps) T ->  Subset-Family of T equals :: DECOMP_1:def 24
 {  B where B is Subset of T : pInt B = psInt B  }  ;
coherence
  {  B where B is Subset of T : pInt B = psInt B  }  is Subset-Family of T
proofend;
func D(sp,ps) T ->  Subset-Family of T equals :: DECOMP_1:def 25
 {  B where B is Subset of T : spInt B = psInt B  }  ;
coherence
  {  B where B is Subset of T : spInt B = psInt B  }  is Subset-Family of T
proofend;
end;

:: deftheorem  defines ^alpha DECOMP_1:def_11_:_
 for T being TopSpace holds T ^alpha =  {  B where B is Subset of T : B is alpha-set of T  }  ;

:: deftheorem  defines SO DECOMP_1:def_12_:_
 for T being TopSpace holds SO T =  {  B where B is Subset of T : B is semi-open  }  ;

:: deftheorem  defines PO DECOMP_1:def_13_:_
 for T being TopSpace holds PO T =  {  B where B is Subset of T : B is pre-open  }  ;

:: deftheorem  defines SPO DECOMP_1:def_14_:_
 for T being TopSpace holds SPO T =  {  B where B is Subset of T : B is semi-pre-open  }  ;

:: deftheorem  defines PSO DECOMP_1:def_15_:_
 for T being TopSpace holds PSO T =  {  B where B is Subset of T : B is pre-semi-open  }  ;

:: deftheorem  defines D(c,alpha) DECOMP_1:def_16_:_
 for T being TopSpace holds D(c,alpha) T =  {  B where B is Subset of T : Int B = alphaInt B  }  ;

:: deftheorem  defines D(c,p) DECOMP_1:def_17_:_
 for T being TopSpace holds D(c,p) T =  {  B where B is Subset of T : Int B = pInt B  }  ;

:: deftheorem  defines D(c,s) DECOMP_1:def_18_:_
 for T being TopSpace holds D(c,s) T =  {  B where B is Subset of T : Int B = sInt B  }  ;

:: deftheorem  defines D(c,ps) DECOMP_1:def_19_:_
 for T being TopSpace holds D(c,ps) T =  {  B where B is Subset of T : Int B = psInt B  }  ;

:: deftheorem  defines D(alpha,p) DECOMP_1:def_20_:_
 for T being TopSpace holds D(alpha,p) T =  {  B where B is Subset of T : alphaInt B = pInt B  }  ;

:: deftheorem  defines D(alpha,s) DECOMP_1:def_21_:_
 for T being TopSpace holds D(alpha,s) T =  {  B where B is Subset of T : alphaInt B = sInt B  }  ;

:: deftheorem  defines D(alpha,ps) DECOMP_1:def_22_:_
 for T being TopSpace holds D(alpha,ps) T =  {  B where B is Subset of T : alphaInt B = psInt B  }  ;

:: deftheorem  defines D(p,sp) DECOMP_1:def_23_:_
 for T being TopSpace holds D(p,sp) T =  {  B where B is Subset of T : pInt B = spInt B  }  ;

:: deftheorem  defines D(p,ps) DECOMP_1:def_24_:_
 for T being TopSpace holds D(p,ps) T =  {  B where B is Subset of T : pInt B = psInt B  }  ;

:: deftheorem  defines D(sp,ps) DECOMP_1:def_25_:_
 for T being TopSpace holds D(sp,ps) T =  {  B where B is Subset of T : spInt B = psInt B  }  ;

theorem Th1: :: DECOMP_1:1
for T being TopSpace
for B being Subset of T holds
( alphaInt B = pInt B iff sInt B = psInt B )
proofend;

theorem Th2: :: DECOMP_1:2
for T being TopSpace
for B being Subset of T holds
( B is alpha-set of T iff B = alphaInt B )
proofend;

theorem Th3: :: DECOMP_1:3
for T being TopSpace
for B being Subset of T holds
( B is semi-open iff B = sInt B )
proofend;

theorem Th4: :: DECOMP_1:4
for T being TopSpace
for B being Subset of T holds
( B is pre-open iff B = pInt B )
proofend;

theorem Th5: :: DECOMP_1:5
for T being TopSpace
for B being Subset of T holds
( B is pre-semi-open iff B = psInt B )
proofend;

theorem Th6: :: DECOMP_1:6
for T being TopSpace
for B being Subset of T holds
( B is semi-pre-open iff B = spInt B )
proofend;

theorem Th7: :: DECOMP_1:7
for T being TopSpace holds (T ^alpha) /\ (D(c,alpha) T) = the topology of T
proofend;

theorem Th8: :: DECOMP_1:8
for T being TopSpace holds (SO T) /\ (D(c,s) T) = the topology of T
proofend;

theorem Th9: :: DECOMP_1:9
for T being TopSpace holds (PO T) /\ (D(c,p) T) = the topology of T
proofend;

theorem Th10: :: DECOMP_1:10
for T being TopSpace holds (PSO T) /\ (D(c,ps) T) = the topology of T
proofend;

theorem Th11: :: DECOMP_1:11
for T being TopSpace holds (PO T) /\ (D(alpha,p) T) = T ^alpha
proofend;

theorem Th12: :: DECOMP_1:12
for T being TopSpace holds (SO T) /\ (D(alpha,s) T) = T ^alpha
proofend;

theorem Th13: :: DECOMP_1:13
for T being TopSpace holds (PSO T) /\ (D(alpha,ps) T) = T ^alpha
proofend;

theorem Th14: :: DECOMP_1:14
for T being TopSpace holds (SPO T) /\ (D(p,sp) T) = PO T
proofend;

theorem Th15: :: DECOMP_1:15
for T being TopSpace holds (PSO T) /\ (D(p,ps) T) = PO T
proofend;

theorem Th16: :: DECOMP_1:16
for T being TopSpace holds (PSO T) /\ (D(alpha,p) T) = SO T
proofend;

theorem Th17: :: DECOMP_1:17
for T being TopSpace holds (PSO T) /\ (D(sp,ps) T) = SPO T
proofend;

definition
let X, Y be non empty TopSpace;
let f be Function of X,Y;
attrf is s-continuous  means :Def26: :: DECOMP_1:def 26
 for G being Subset of Y st G is open holds
f " G in SO X;
attrf is p-continuous  means :Def27: :: DECOMP_1:def 27
 for G being Subset of Y st G is open holds
f " G in PO X;
attrf is alpha-continuous  means :Def28: :: DECOMP_1:def 28
 for G being Subset of Y st G is open holds
f " G in X ^alpha ;
attrf is ps-continuous  means :Def29: :: DECOMP_1:def 29
 for G being Subset of Y st G is open holds
f " G in PSO X;
attrf is sp-continuous  means :Def30: :: DECOMP_1:def 30
 for G being Subset of Y st G is open holds
f " G in SPO X;
attrf is (c,alpha)-continuous  means :Def31: :: DECOMP_1:def 31
 for G being Subset of Y st G is open holds
f " G in D(c,alpha) X;
attrf is (c,s)-continuous  means :Def32: :: DECOMP_1:def 32
 for G being Subset of Y st G is open holds
f " G in D(c,s) X;
attrf is (c,p)-continuous  means :Def33: :: DECOMP_1:def 33
 for G being Subset of Y st G is open holds
f " G in D(c,p) X;
attrf is (c,ps)-continuous  means :Def34: :: DECOMP_1:def 34
 for G being Subset of Y st G is open holds
f " G in D(c,ps) X;
attrf is (alpha,p)-continuous  means :Def35: :: DECOMP_1:def 35
 for G being Subset of Y st G is open holds
f " G in D(alpha,p) X;
attrf is (alpha,s)-continuous  means :Def36: :: DECOMP_1:def 36
 for G being Subset of Y st G is open holds
f " G in D(alpha,s) X;
attrf is (alpha,ps)-continuous  means :Def37: :: DECOMP_1:def 37
 for G being Subset of Y st G is open holds
f " G in D(alpha,ps) X;
attrf is (p,ps)-continuous  means :: DECOMP_1:def 38
 for G being Subset of Y st G is open holds
f " G in D(p,ps) X;
attrf is (p,sp)-continuous  means :: DECOMP_1:def 39
 for G being Subset of Y st G is open holds
f " G in D(p,sp) X;
attrf is (sp,ps)-continuous  means :: DECOMP_1:def 40
 for G being Subset of Y st G is open holds
f " G in D(sp,ps) X;
end;

:: deftheorem Def26 defines s-continuous DECOMP_1:def_26_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is s-continuous iff for G being Subset of Y st G is open holds
f " G in SO X );

:: deftheorem Def27 defines p-continuous DECOMP_1:def_27_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is p-continuous iff for G being Subset of Y st G is open holds
f " G in PO X );

:: deftheorem Def28 defines alpha-continuous DECOMP_1:def_28_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is alpha-continuous iff for G being Subset of Y st G is open holds
f " G in X ^alpha );

:: deftheorem Def29 defines ps-continuous DECOMP_1:def_29_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is ps-continuous iff for G being Subset of Y st G is open holds
f " G in PSO X );

:: deftheorem Def30 defines sp-continuous DECOMP_1:def_30_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is sp-continuous iff for G being Subset of Y st G is open holds
f " G in SPO X );

:: deftheorem Def31 defines (c,alpha)-continuous DECOMP_1:def_31_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (c,alpha)-continuous iff for G being Subset of Y st G is open holds
f " G in D(c,alpha) X );

:: deftheorem Def32 defines (c,s)-continuous DECOMP_1:def_32_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (c,s)-continuous iff for G being Subset of Y st G is open holds
f " G in D(c,s) X );

:: deftheorem Def33 defines (c,p)-continuous DECOMP_1:def_33_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (c,p)-continuous iff for G being Subset of Y st G is open holds
f " G in D(c,p) X );

:: deftheorem Def34 defines (c,ps)-continuous DECOMP_1:def_34_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (c,ps)-continuous iff for G being Subset of Y st G is open holds
f " G in D(c,ps) X );

:: deftheorem Def35 defines (alpha,p)-continuous DECOMP_1:def_35_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (alpha,p)-continuous iff for G being Subset of Y st G is open holds
f " G in D(alpha,p) X );

:: deftheorem Def36 defines (alpha,s)-continuous DECOMP_1:def_36_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (alpha,s)-continuous iff for G being Subset of Y st G is open holds
f " G in D(alpha,s) X );

:: deftheorem Def37 defines (alpha,ps)-continuous DECOMP_1:def_37_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (alpha,ps)-continuous iff for G being Subset of Y st G is open holds
f " G in D(alpha,ps) X );

:: deftheorem  defines (p,ps)-continuous DECOMP_1:def_38_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (p,ps)-continuous iff for G being Subset of Y st G is open holds
f " G in D(p,ps) X );

:: deftheorem  defines (p,sp)-continuous DECOMP_1:def_39_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (p,sp)-continuous iff for G being Subset of Y st G is open holds
f " G in D(p,sp) X );

:: deftheorem  defines (sp,ps)-continuous DECOMP_1:def_40_:_
 for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is (sp,ps)-continuous iff for G being Subset of Y st G is open holds
f " G in D(sp,ps) X );

theorem :: DECOMP_1:18
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is alpha-continuous iff ( f is p-continuous & f is (alpha,p)-continuous ) )
proofend;

theorem :: DECOMP_1:19
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is alpha-continuous iff ( f is s-continuous & f is (alpha,s)-continuous ) )
proofend;

theorem :: DECOMP_1:20
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is alpha-continuous iff ( f is ps-continuous & f is (alpha,ps)-continuous ) )
proofend;

theorem :: DECOMP_1:21
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is p-continuous iff ( f is sp-continuous & f is (p,sp)-continuous ) )
proofend;

theorem :: DECOMP_1:22
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is p-continuous iff ( f is ps-continuous & f is (p,ps)-continuous ) )
proofend;

theorem :: DECOMP_1:23
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is s-continuous iff ( f is ps-continuous & f is (alpha,p)-continuous ) )
proofend;

theorem :: DECOMP_1:24
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is sp-continuous iff ( f is ps-continuous & f is (sp,ps)-continuous ) )
proofend;

theorem :: DECOMP_1:25
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is continuous iff ( f is alpha-continuous & f is (c,alpha)-continuous ) )
proofend;

theorem :: DECOMP_1:26
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is continuous iff ( f is s-continuous & f is (c,s)-continuous ) )
proofend;

theorem :: DECOMP_1:27
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is continuous iff ( f is p-continuous & f is (c,p)-continuous ) )
proofend;

theorem :: DECOMP_1:28
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is continuous iff ( f is ps-continuous & f is (c,ps)-continuous ) )
proofend;