Article COMPACT1, MML version 4.99.1005

:: COMPACT1:attrnot 1 => COMPACT1:attr 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is compact means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is compact(a1);
end;

:: COMPACT1:dfs 1
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is compact
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is compact(a1);

:: COMPACT1:def 1
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is compact(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is compact(b1);

:: COMPACT1:attrnot 2 => COMPACT1:attr 2
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is relatively-compact means
    Cl a2 is compact(a1);
end;

:: COMPACT1:dfs 2
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is relatively-compact
it is sufficient to prove
  thus Cl a2 is compact(a1);

:: COMPACT1:def 2
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is relatively-compact(b1)
   iff
      Cl b2 is compact(b1);

:: COMPACT1:funcreg 1
registration
  let a1 be TopSpace-like TopStruct;
  cluster {} a1 -> relatively-compact;
end;

:: COMPACT1:exreg 1
registration
  let a1 be TopSpace-like TopStruct;
  cluster relatively-compact Element of bool the carrier of a1;
end;

:: COMPACT1:funcreg 2
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be relatively-compact Element of bool the carrier of a1;
  cluster Cl a2 -> compact;
end;

:: COMPACT1:attrnot 3 => COMPACT1:attr 2
notation
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  synonym pre-compact for relatively-compact;
end;

:: COMPACT1:attrnot 4 => WAYBEL_3:attr 6
notation
  let a1 be non empty TopSpace-like TopStruct;
  synonym liminally-compact for locally-compact;
end;

:: COMPACT1:attrnot 5 => WAYBEL_3:attr 6
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is liminally-compact means
    for b1 being Element of the carrier of a1 holds
       ex b2 being basis of b1 st
          b2 is compact(a1);
end;

:: COMPACT1:dfs 3
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is locally-compact
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being basis of b1 st
          b2 is compact(a1);

:: COMPACT1:def 3
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is locally-compact
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being basis of b2 st
            b3 is compact(b1);

:: COMPACT1:attrnot 6 => COMPACT1:attr 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is locally-relatively-compact means
    for b1 being Element of the carrier of a1 holds
       ex b2 being a_neighborhood of b1 st
          b2 is relatively-compact(a1);
end;

:: COMPACT1:dfs 4
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is locally-relatively-compact
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being a_neighborhood of b1 st
          b2 is relatively-compact(a1);

:: COMPACT1:def 4
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is locally-relatively-compact
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being a_neighborhood of b2 st
            b3 is relatively-compact(b1);

:: COMPACT1:attrnot 7 => COMPACT1:attr 4
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is locally-closed/compact means
    for b1 being Element of the carrier of a1 holds
       ex b2 being a_neighborhood of b1 st
          b2 is closed(a1) & b2 is compact(a1);
end;

:: COMPACT1:dfs 5
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is locally-closed/compact
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being a_neighborhood of b1 st
          b2 is closed(a1) & b2 is compact(a1);

:: COMPACT1:def 5
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is locally-closed/compact
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being a_neighborhood of b2 st
            b3 is closed(b1) & b3 is compact(b1);

:: COMPACT1:attrnot 8 => COMPACT1:attr 5
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is locally-compact means
    for b1 being Element of the carrier of a1 holds
       ex b2 being a_neighborhood of b1 st
          b2 is compact(a1);
end;

:: COMPACT1:dfs 6
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is locally-compact
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being a_neighborhood of b1 st
          b2 is compact(a1);

:: COMPACT1:def 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is locally-compact
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being a_neighborhood of b2 st
            b3 is compact(b1);

:: COMPACT1:condreg 1
registration
  cluster non empty TopSpace-like locally-compact -> locally-compact (TopStruct);
end;

:: COMPACT1:condreg 2
registration
  cluster non empty TopSpace-like being_T3 locally-compact -> locally-compact (TopStruct);
end;

:: COMPACT1:condreg 3
registration
  cluster non empty TopSpace-like locally-relatively-compact -> locally-closed/compact (TopStruct);
end;

:: COMPACT1:condreg 4
registration
  cluster non empty TopSpace-like locally-closed/compact -> locally-relatively-compact (TopStruct);
end;

:: COMPACT1:condreg 5
registration
  cluster non empty TopSpace-like locally-relatively-compact -> locally-compact (TopStruct);
end;

:: COMPACT1:condreg 6
registration
  cluster non empty TopSpace-like being_T2 locally-compact -> locally-relatively-compact (TopStruct);
end;

:: COMPACT1:condreg 7
registration
  cluster non empty TopSpace-like compact -> locally-compact (TopStruct);
end;

:: COMPACT1:condreg 8
registration
  cluster non empty TopSpace-like discrete -> locally-compact (TopStruct);
end;

:: COMPACT1:exreg 2
registration
  cluster non empty TopSpace-like discrete TopStruct;
end;

:: COMPACT1:funcreg 3
registration
  let a1 be non empty TopSpace-like locally-compact TopStruct;
  let a2 be non empty closed Element of bool the carrier of a1;
  cluster a1 | a2 -> strict locally-compact;
end;

:: COMPACT1:funcreg 4
registration
  let a1 be non empty TopSpace-like being_T3 locally-compact TopStruct;
  let a2 be non empty open Element of bool the carrier of a1;
  cluster a1 | a2 -> strict locally-compact;
end;

:: COMPACT1:th 1
theorem
for b1 being non empty TopSpace-like being_T2 TopStruct
for b2 being non empty Element of bool the carrier of b1
      st b1 | b2 is dense(b1) & b1 | b2 is locally-compact
   holds b1 | b2 is open(b1);

:: COMPACT1:th 2
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
      st [#] b1 c= [#] b2
   holds (incl(b1,b2)) .: b3 = b3;

:: COMPACT1:attrnot 9 => COMPACT1:attr 6
definition
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is embedding means
    ex b1 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 | rng a3 st
       b1 = a3 & b1 is being_homeomorphism(a1, a2 | rng a3);
end;

:: COMPACT1:dfs 7
definiens
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is embedding
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 | rng a3 st
       b1 = a3 & b1 is being_homeomorphism(a1, a2 | rng a3);

:: COMPACT1:def 7
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is embedding(b1, b2)
   iff
      ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 | rng b3 st
         b4 = b3 & b4 is being_homeomorphism(b1, b2 | rng b3);

:: COMPACT1:th 3
theorem
for b1, b2 being TopSpace-like TopStruct
      st [#] b1 c= [#] b2 &
         (ex b3 being Element of bool the carrier of b2 st
            b3 = [#] b1 & the topology of b2 | b3 = the topology of b1)
   holds incl(b1,b2) is embedding(b1, b2);

:: COMPACT1:attrnot 10 => COMPACT1:attr 7
definition
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is compactification means
    a3 is embedding(a1, a2) & a2 is compact & a3 .: [#] a1 is dense(a2);
end;

:: COMPACT1:dfs 8
definiens
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is compactification
it is sufficient to prove
  thus a3 is embedding(a1, a2) & a2 is compact & a3 .: [#] a1 is dense(a2);

:: COMPACT1:def 8
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is compactification(b1, b2)
   iff
      b3 is embedding(b1, b2) & b2 is compact & b3 .: [#] b1 is dense(b2);

:: COMPACT1:condreg 9
registration
  let a1, a2 be TopSpace-like TopStruct;
  cluster Function-like quasi_total compactification -> embedding (Relation of the carrier of a1,the carrier of a2);
end;

:: COMPACT1:funcnot 1 => COMPACT1:func 1
definition
  let a1 be TopStruct;
  func One-Point_Compactification A1 -> strict TopStruct means
    the carrier of it = succ [#] a1 &
     the topology of it = (the topology of a1) \/ {b1 \/ {[#] a1} where b1 is Element of bool the carrier of a1: b1 is open(a1) & b1 ` is compact(a1)};
end;

:: COMPACT1:def 9
theorem
for b1 being TopStruct
for b2 being strict TopStruct holds
      b2 = One-Point_Compactification b1
   iff
      the carrier of b2 = succ [#] b1 &
       the topology of b2 = (the topology of b1) \/ {b3 \/ {[#] b1} where b3 is Element of bool the carrier of b1: b3 is open(b1) & b3 ` is compact(b1)};

:: COMPACT1:funcreg 5
registration
  let a1 be TopStruct;
  cluster One-Point_Compactification a1 -> non empty strict;
end;

:: COMPACT1:th 4
theorem
for b1 being TopStruct holds
   [#] b1 c= [#] One-Point_Compactification b1;

:: COMPACT1:funcreg 6
registration
  let a1 be TopSpace-like TopStruct;
  cluster One-Point_Compactification a1 -> strict TopSpace-like;
end;

:: COMPACT1:th 5
theorem
for b1 being TopStruct holds
   b1 is SubSpace of One-Point_Compactification b1;

:: COMPACT1:funcreg 7
registration
  let a1 be TopSpace-like TopStruct;
  cluster One-Point_Compactification a1 -> strict compact;
end;

:: COMPACT1:th 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is being_T2 & b1 is locally-compact
   iff
      One-Point_Compactification b1 is being_T2;

:: COMPACT1:th 7
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is not compact
   iff
      ex b2 being Element of bool the carrier of One-Point_Compactification b1 st
         b2 = [#] b1 & b2 is dense(One-Point_Compactification b1);

:: COMPACT1:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is not compact
   holds incl(b1,One-Point_Compactification b1) is compactification(b1, One-Point_Compactification b1);