Article FINTOPO4, MML version 4.99.1005

:: FINTOPO4:prednot 1 => FINTOPO4:pred 1
definition
  let a1 be non empty RelStr;
  let a2, a3 be Element of bool the carrier of a1;
  pred A2,A3 are_separated means
    a2 ^b misses a3 & a2 misses a3 ^b;
  symmetry;
::  for a1 being non empty RelStr
::  for a2, a3 being Element of bool the carrier of a1
::        st a2,a3 are_separated
::     holds a3,a2 are_separated;
end;

:: FINTOPO4:dfs 1
definiens
  let a1 be non empty RelStr;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a2,a3 are_separated
it is sufficient to prove
  thus a2 ^b misses a3 & a2 misses a3 ^b;

:: FINTOPO4:def 1
theorem
for b1 being non empty RelStr
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_separated
iff
   b2 ^b misses b3 & b2 misses b3 ^b;

:: FINTOPO4:th 1
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
      st b3 <= b4
   holds Finf(b2,b3) c= Finf(b2,b4);

:: FINTOPO4:th 2
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
      st b3 <= b4
   holds Fcl(b2,b3) c= Fcl(b2,b4);

:: FINTOPO4:th 3
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
      st b3 <= b4
   holds Fdfl(b2,b4) c= Fdfl(b2,b3);

:: FINTOPO4:th 4
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of NAT
      st b3 <= b4
   holds Fint(b2,b4) c= Fint(b2,b3);

:: FINTOPO4:th 5
theorem
for b1 being non empty RelStr
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 are_separated
   holds b3,b2 are_separated;

:: FINTOPO4:th 6
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 are_separated
   holds b2 misses b3;

:: FINTOPO4:th 7
theorem
for b1 being non empty RelStr
for b2, b3 being Element of bool the carrier of b1
      st b1 is symmetric
   holds    b2,b3 are_separated
   iff
      b2 ^f misses b3 & b2 misses b3 ^f;

:: FINTOPO4:th 8
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
      st b1 is symmetric & b2 ^b misses b3
   holds b2 misses b3 ^b;

:: FINTOPO4:th 9
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
      st b1 is symmetric & b2 misses b3 ^b
   holds b2 ^b misses b3;

:: FINTOPO4:th 10
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
      st b1 is symmetric
   holds    b2,b3 are_separated
   iff
      b2 ^b misses b3;

:: FINTOPO4:th 11
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
      st b1 is symmetric
   holds    b2,b3 are_separated
   iff
      b2 misses b3 ^b;

:: FINTOPO4:th 12
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
      st b1 is symmetric
   holds    b2 is connected(b1)
   iff
      for b3, b4 being Element of bool the carrier of b1
            st b2 = b3 \/ b4 & b3,b4 are_separated & b3 <> b2
         holds b4 = b2;

:: FINTOPO4:th 13
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
      st b1 is symmetric
   holds    b2 is connected(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 <> {} & b2 \ b3 <> {} & b3 c= b2
         holds b3 ^b meets b2 \ b3;

:: FINTOPO4:prednot 2 => FINTOPO4:pred 2
definition
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  let a4 be Element of NAT;
  pred A3 is_continuous A4 means
    for b1 being Element of the carrier of a1
    for b2 being Element of the carrier of a2
          st b1 in the carrier of a1 & b2 = a3 . b1
       holds a3 .: U_FT(b1,0) c= U_FT(b2,a4);
end;

:: FINTOPO4:dfs 2
definiens
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  let a4 be Element of NAT;
To prove
     a3 is_continuous a4
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
    for b2 being Element of the carrier of a2
          st b1 in the carrier of a1 & b2 = a3 . b1
       holds a3 .: U_FT(b1,0) c= U_FT(b2,a4);

:: FINTOPO4:def 2
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 holds
      b3 is_continuous b4
   iff
      for b5 being Element of the carrier of b1
      for b6 being Element of the carrier of b2
            st b5 in the carrier of b1 & b6 = b3 . b5
         holds b3 .: U_FT(b5,0) c= U_FT(b6,b4);

:: FINTOPO4:th 14
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive RelStr
for b3 being Element of NAT
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is_continuous 0
   holds b4 is_continuous b3;

:: FINTOPO4:th 15
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive RelStr
for b3, b4 being Element of NAT
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b5 is_continuous b3 & b3 <= b4
   holds b5 is_continuous b4;

:: FINTOPO4:th 16
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b5 is_continuous 0 & b4 = b5 .: b3
   holds b5 .: (b3 ^b) c= b4 ^b;

:: FINTOPO4:th 17
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is connected(b1) & b5 is_continuous 0 & b4 = b5 .: b3
   holds b4 is connected(b2);

:: FINTOPO4:funcnot 1 => FINTOPO4:func 1
definition
  let a1 be natural set;
  func Nbdl1 A1 -> Relation of Seg a1,Seg a1 means
    for b1 being Element of NAT
          st b1 in Seg a1
       holds Im(it,b1) = {b1,max(b1 -' 1,1),min(b1 + 1,a1)};
end;

:: FINTOPO4:def 3
theorem
for b1 being natural set
for b2 being Relation of Seg b1,Seg b1 holds
      b2 = Nbdl1 b1
   iff
      for b3 being Element of NAT
            st b3 in Seg b1
         holds Im(b2,b3) = {b3,max(b3 -' 1,1),min(b3 + 1,b1)};

:: FINTOPO4:funcnot 2 => FINTOPO4:func 2
definition
  let a1 be natural set;
  assume 0 < a1;
  func FTSL1 A1 -> non empty RelStr equals
    RelStr(#Seg a1,Nbdl1 a1#);
end;

:: FINTOPO4:def 4
theorem
for b1 being natural set
      st 0 < b1
   holds FTSL1 b1 = RelStr(#Seg b1,Nbdl1 b1#);

:: FINTOPO4:th 18
theorem
for b1 being natural set
      st 0 < b1
   holds FTSL1 b1 is reflexive;

:: FINTOPO4:th 19
theorem
for b1 being natural set
      st 0 < b1
   holds FTSL1 b1 is symmetric;

:: FINTOPO4:funcnot 3 => FINTOPO4:func 3
definition
  let a1 be natural set;
  func Nbdc1 A1 -> Relation of Seg a1,Seg a1 means
    for b1 being Element of NAT
          st b1 in Seg a1
       holds (1 < b1 & b1 < a1 implies Im(it,b1) = {b1,b1 -' 1,b1 + 1}) &
        (b1 = 1 & b1 < a1 implies Im(it,b1) = {b1,a1,b1 + 1}) &
        (1 < b1 & b1 = a1 implies Im(it,b1) = {b1,b1 -' 1,1}) &
        (b1 = 1 & b1 = a1 implies Im(it,b1) = {b1});
end;

:: FINTOPO4:def 5
theorem
for b1 being natural set
for b2 being Relation of Seg b1,Seg b1 holds
      b2 = Nbdc1 b1
   iff
      for b3 being Element of NAT
            st b3 in Seg b1
         holds (1 < b3 & b3 < b1 implies Im(b2,b3) = {b3,b3 -' 1,b3 + 1}) &
          (b3 = 1 & b3 < b1 implies Im(b2,b3) = {b3,b1,b3 + 1}) &
          (1 < b3 & b3 = b1 implies Im(b2,b3) = {b3,b3 -' 1,1}) &
          (b3 = 1 & b3 = b1 implies Im(b2,b3) = {b3});

:: FINTOPO4:funcnot 4 => FINTOPO4:func 4
definition
  let a1 be natural set;
  assume 0 < a1;
  func FTSC1 A1 -> non empty RelStr equals
    RelStr(#Seg a1,Nbdc1 a1#);
end;

:: FINTOPO4:def 6
theorem
for b1 being natural set
      st 0 < b1
   holds FTSC1 b1 = RelStr(#Seg b1,Nbdc1 b1#);

:: FINTOPO4:th 20
theorem
for b1 being Element of NAT
      st 0 < b1
   holds FTSC1 b1 is reflexive;

:: FINTOPO4:th 21
theorem
for b1 being Element of NAT
      st 0 < b1
   holds FTSC1 b1 is symmetric;