Article FINTOPO5, MML version 4.99.1005

:: FINTOPO5:th 1
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b1
      st b3 is one-to-one
   holds b3 " .: (b3 .: b4) = b4;

:: FINTOPO5:th 2
theorem
for b1 being Element of NAT holds
      0 < b1
   iff
      Seg b1 <> {};

:: FINTOPO5:attrnot 1 => FINTOPO5:attr 1
definition
  let a1, a2 be RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is being_homeomorphism means
    a3 is one-to-one &
     a3 is onto(the carrier of a1, the carrier of a2) &
     (for b1 being Element of the carrier of a1 holds
        a3 .: U_FT b1 = Im(the InternalRel of a2,a3 . b1));
end;

:: FINTOPO5:dfs 1
definiens
  let a1, a2 be RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is being_homeomorphism
it is sufficient to prove
  thus a3 is one-to-one &
     a3 is onto(the carrier of a1, the carrier of a2) &
     (for b1 being Element of the carrier of a1 holds
        a3 .: U_FT b1 = Im(the InternalRel of a2,a3 . b1));

:: FINTOPO5:def 1
theorem
for b1, b2 being RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_homeomorphism(b1, b2)
   iff
      b3 is one-to-one &
       b3 is onto(the carrier of b1, the carrier of b2) &
       (for b4 being Element of the carrier of b1 holds
          b3 .: U_FT b4 = Im(the InternalRel of b2,b3 . b4));

:: FINTOPO5:th 3
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is being_homeomorphism(b1, b2)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
      b4 = b3 " & b4 is being_homeomorphism(b2, b1);

:: FINTOPO5:th 4
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of NAT
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
   st b3 is being_homeomorphism(b1, b2) & b6 = b3 . b5
for b7 being Element of the carrier of b1 holds
      b7 in U_FT(b5,b4)
   iff
      b3 . b7 in U_FT(b6,b4);

:: FINTOPO5:th 5
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of NAT
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
   st b3 is being_homeomorphism(b1, b2) & b6 = b3 . b5
for b7 being Element of the carrier of b2 holds
      b3 " . b7 in U_FT(b5,b4)
   iff
      b7 in U_FT(b6,b4);

:: FINTOPO5:th 6
theorem
for b1 being non empty Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of FTSL1 b1,the carrier of FTSL1 b1
      st b2 is_continuous 0
   holds ex b3 being Element of the carrier of FTSL1 b1 st
      b2 . b3 in U_FT(b3,0);

:: FINTOPO5:th 7
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT
      st b1 is reflexive
   holds U_FT(b2,b3) c= U_FT(b2,b3 + 1);

:: FINTOPO5:th 8
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT
      st b1 is reflexive
   holds U_FT(b2,0) c= U_FT(b2,b3);

:: FINTOPO5:th 9
theorem
for b1 being non empty natural set
for b2, b3, b4 being natural set
for b5 being Element of the carrier of FTSL1 b1
      st b5 = b2
   holds    b3 in U_FT(b5,b4)
   iff
      b3 in Seg b1 & abs (b2 - b3) <= b4 + 1;

:: FINTOPO5:th 10
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of NAT
for b4 being Function-like quasi_total Relation of the carrier of FTSL1 b3,the carrier of FTSL1 b3
      st b4 is_continuous b1 & b2 = [/b1 / 2\]
   holds ex b5 being Element of the carrier of FTSL1 b3 st
      b4 . b5 in U_FT(b5,b2);

:: FINTOPO5:funcnot 1 => FINTOPO5:func 1
definition
  let a1, a2 be set;
  let a3 be Relation of a1,a2;
  let a4 be set;
  redefine func Im(a3,a4) -> Element of bool a2;
end;

:: FINTOPO5:funcnot 2 => FINTOPO5:func 2
definition
  let a1, a2 be Element of NAT;
  func Nbdl2(A1,A2) -> Relation of [:Seg a1,Seg a2:],[:Seg a1,Seg a2:] means
    for b1 being set
       st b1 in [:Seg a1,Seg a2:]
    for b2, b3 being Element of NAT
          st b1 = [b2,b3]
       holds Im(it,b1) = [:Im(Nbdl1 a1,b2),Im(Nbdl1 a2,b3):];
end;

:: FINTOPO5:def 2
theorem
for b1, b2 being Element of NAT
for b3 being Relation of [:Seg b1,Seg b2:],[:Seg b1,Seg b2:] holds
      b3 = Nbdl2(b1,b2)
   iff
      for b4 being set
         st b4 in [:Seg b1,Seg b2:]
      for b5, b6 being Element of NAT
            st b4 = [b5,b6]
         holds Im(b3,b4) = [:Im(Nbdl1 b1,b5),Im(Nbdl1 b2,b6):];

:: FINTOPO5:funcnot 3 => FINTOPO5:func 3
definition
  let a1, a2 be Element of NAT;
  func FTSL2(A1,A2) -> strict RelStr equals
    RelStr(#[:Seg a1,Seg a2:],Nbdl2(a1,a2)#);
end;

:: FINTOPO5:def 3
theorem
for b1, b2 being Element of NAT holds
FTSL2(b1,b2) = RelStr(#[:Seg b1,Seg b2:],Nbdl2(b1,b2)#);

:: FINTOPO5:funcreg 1
registration
  let a1, a2 be non empty Element of NAT;
  cluster FTSL2(a1,a2) -> non empty strict;
end;

:: FINTOPO5:th 11
theorem
for b1, b2 being non empty Element of NAT holds
FTSL2(b1,b2) is reflexive;

:: FINTOPO5:th 12
theorem
for b1, b2 being non empty Element of NAT holds
FTSL2(b1,b2) is symmetric;

:: FINTOPO5:th 13
theorem
for b1 being non empty Element of NAT holds
   ex b2 being Function-like quasi_total Relation of the carrier of FTSL2(b1,1),the carrier of FTSL1 b1 st
      b2 is being_homeomorphism(FTSL2(b1,1), FTSL1 b1);

:: FINTOPO5:funcnot 4 => FINTOPO5:func 4
definition
  let a1, a2 be Element of NAT;
  func Nbds2(A1,A2) -> Relation of [:Seg a1,Seg a2:],[:Seg a1,Seg a2:] means
    for b1 being set
       st b1 in [:Seg a1,Seg a2:]
    for b2, b3 being Element of NAT
          st b1 = [b2,b3]
       holds Im(it,b1) = [:{b2},Im(Nbdl1 a2,b3):] \/ [:Im(Nbdl1 a1,b2),{b3}:];
end;

:: FINTOPO5:def 4
theorem
for b1, b2 being Element of NAT
for b3 being Relation of [:Seg b1,Seg b2:],[:Seg b1,Seg b2:] holds
      b3 = Nbds2(b1,b2)
   iff
      for b4 being set
         st b4 in [:Seg b1,Seg b2:]
      for b5, b6 being Element of NAT
            st b4 = [b5,b6]
         holds Im(b3,b4) = [:{b5},Im(Nbdl1 b2,b6):] \/ [:Im(Nbdl1 b1,b5),{b6}:];

:: FINTOPO5:funcnot 5 => FINTOPO5:func 5
definition
  let a1, a2 be Element of NAT;
  func FTSS2(A1,A2) -> strict RelStr equals
    RelStr(#[:Seg a1,Seg a2:],Nbds2(a1,a2)#);
end;

:: FINTOPO5:def 5
theorem
for b1, b2 being Element of NAT holds
FTSS2(b1,b2) = RelStr(#[:Seg b1,Seg b2:],Nbds2(b1,b2)#);

:: FINTOPO5:funcreg 2
registration
  let a1, a2 be non empty Element of NAT;
  cluster FTSS2(a1,a2) -> non empty strict;
end;

:: FINTOPO5:th 14
theorem
for b1, b2 being non empty Element of NAT holds
FTSS2(b1,b2) is reflexive;

:: FINTOPO5:th 15
theorem
for b1, b2 being non empty Element of NAT holds
FTSS2(b1,b2) is symmetric;

:: FINTOPO5:th 16
theorem
for b1 being non empty Element of NAT holds
   ex b2 being Function-like quasi_total Relation of the carrier of FTSS2(b1,1),the carrier of FTSL1 b1 st
      b2 is being_homeomorphism(FTSS2(b1,1), FTSL1 b1);