Article YELLOW13, MML version 4.99.1005
:: YELLOW13:exreg 1
registration
cluster non empty trivial strict RelStr;
end;
:: YELLOW13:exreg 2
registration
cluster non empty trivial strict TopRelStr;
end;
:: YELLOW13:th 1
theorem
for b1 being non empty TopSpace-like being_T1 TopStruct
for b2 being finite Element of bool the carrier of b1 holds
b2 is closed(b1);
:: YELLOW13:condreg 1
registration
let a1 be non empty TopSpace-like being_T1 TopStruct;
cluster finite -> closed (Element of bool the carrier of a1);
end;
:: YELLOW13:funcreg 1
registration
let a1 be compact TopStruct;
cluster [#] a1 -> compact;
end;
:: YELLOW13:exreg 3
registration
cluster non empty trivial strict TopSpace-like TopStruct;
end;
:: YELLOW13:condreg 2
registration
cluster non empty finite TopSpace-like being_T1 -> discrete (TopStruct);
end;
:: YELLOW13:condreg 3
registration
cluster finite TopSpace-like -> compact (TopStruct);
end;
:: YELLOW13:th 2
theorem
for b1 being non empty TopSpace-like discrete TopStruct holds
b1 is being_T4;
:: YELLOW13:th 3
theorem
for b1 being non empty TopSpace-like discrete TopStruct holds
b1 is being_T3;
:: YELLOW13:th 4
theorem
for b1 being non empty TopSpace-like discrete TopStruct holds
b1 is being_T2;
:: YELLOW13:th 5
theorem
for b1 being non empty TopSpace-like discrete TopStruct holds
b1 is being_T1;
:: YELLOW13:condreg 4
registration
cluster non empty TopSpace-like discrete -> being_T1 being_T2 being_T3 being_T4 (TopStruct);
end;
:: YELLOW13:condreg 5
registration
cluster non empty TopSpace-like being_T1 being_T4 -> being_T3 (TopStruct);
end;
:: YELLOW13:condreg 6
registration
cluster TopSpace-like being_T1 being_T3 -> being_T2 (TopStruct);
end;
:: YELLOW13:condreg 7
registration
cluster TopSpace-like being_T2 -> being_T1 (TopStruct);
end;
:: YELLOW13:condreg 8
registration
cluster TopSpace-like being_T1 -> discerning (TopStruct);
end;
:: YELLOW13:th 6
theorem
for b1 being reflexive RelStr
for b2 being reflexive transitive RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1 holds
downarrow (b3 .: b4) c= downarrow (b3 .: downarrow b4);
:: YELLOW13:th 7
theorem
for b1 being reflexive RelStr
for b2 being reflexive transitive RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
st b3 is monotone(b1, b2)
holds downarrow (b3 .: b4) = downarrow (b3 .: downarrow b4);
:: YELLOW13:th 8
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr holds
IdsMap b1 is one-to-one;
:: YELLOW13:funcreg 2
registration
let a1 be non empty reflexive transitive antisymmetric RelStr;
cluster IdsMap a1 -> Function-like one-to-one quasi_total;
end;
:: YELLOW13:th 9
theorem
for b1 being finite reflexive transitive antisymmetric with_suprema with_infima RelStr holds
SupMap b1 is one-to-one;
:: YELLOW13:funcreg 3
registration
let a1 be finite reflexive transitive antisymmetric with_suprema with_infima RelStr;
cluster SupMap a1 -> Function-like one-to-one quasi_total;
end;
:: YELLOW13:th 10
theorem
for b1 being finite reflexive transitive antisymmetric with_suprema with_infima RelStr holds
b1,InclPoset Ids b1 are_isomorphic;
:: YELLOW13:th 11
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being Element of the carrier of b1
for b3 being non empty Element of bool the carrier of b1 holds
b2 "/\" preserves_inf_of b3;
:: YELLOW13:th 12
theorem
for b1 being non empty reflexive transitive antisymmetric complete RelStr
for b2 being Element of the carrier of b1 holds
b2 "/\" is meet-preserving(b1, b1);
:: YELLOW13:funcreg 4
registration
let a1 be non empty reflexive transitive antisymmetric complete RelStr;
let a2 be Element of the carrier of a1;
cluster a2 "/\" -> Function-like quasi_total meet-preserving;
end;
:: YELLOW13:th 13
theorem
for b1 being non empty anti-discrete TopStruct
for b2 being Element of the carrier of b1 holds
{the carrier of b1} is Basis of b2;
:: YELLOW13:th 14
theorem
for b1 being non empty anti-discrete TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2 holds
b3 = {the carrier of b1};
:: YELLOW13:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Basis of b1
for b3 being Element of the carrier of b1 holds
{b4 where b4 is Element of bool the carrier of b1: b4 in b2 & b3 in b4} is Basis of b3;
:: YELLOW13:th 16
theorem
for b1 being non empty TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in Cl b2
iff
for b4 being Basis of b3
for b5 being Element of bool the carrier of b1
st b5 in b4
holds b2 meets b5;
:: YELLOW13:th 17
theorem
for b1 being non empty TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in Cl b2
iff
ex b4 being Basis of b3 st
for b5 being Element of bool the carrier of b1
st b5 in b4
holds b2 meets b5;
:: YELLOW13:modenot 1 => YELLOW13:mode 1
definition
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
mode basis of A2 -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1
st a2 in Int b1
holds ex b2 being Element of bool the carrier of a1 st
b2 in it & a2 in Int b2 & b2 c= b1;
end;
:: YELLOW13:dfs 1
definiens
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
let a3 be Element of bool bool the carrier of a1;
To prove
a3 is basis of a2
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st a2 in Int b1
holds ex b2 being Element of bool the carrier of a1 st
b2 in a3 & a2 in Int b2 & b2 c= b1;
:: YELLOW13:def 1
theorem
for b1 being TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
b3 is basis of b2
iff
for b4 being Element of bool the carrier of b1
st b2 in Int b4
holds ex b5 being Element of bool the carrier of b1 st
b5 in b3 & b2 in Int b5 & b5 c= b4;
:: YELLOW13:modenot 2 => YELLOW13:mode 1
definition
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
mode basis of A2 -> Element of bool bool the carrier of a1 means
for b1 being a_neighborhood of a2 holds
ex b2 being a_neighborhood of a2 st
b2 in it & b2 c= b1;
end;
:: YELLOW13:dfs 2
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
let a3 be Element of bool bool the carrier of a1;
To prove
a3 is basis of a2
it is sufficient to prove
thus for b1 being a_neighborhood of a2 holds
ex b2 being a_neighborhood of a2 st
b2 in a3 & b2 c= b1;
:: YELLOW13:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
b3 is basis of b2
iff
for b4 being a_neighborhood of b2 holds
ex b5 being a_neighborhood of b2 st
b5 in b3 & b5 c= b4;
:: YELLOW13:th 18
theorem
for b1 being TopStruct
for b2 being Element of the carrier of b1 holds
bool the carrier of b1 is basis of b2;
:: YELLOW13:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being basis of b2 holds
b3 is not empty;
:: YELLOW13:condreg 9
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster -> non empty (basis of a2);
end;
:: YELLOW13:exreg 4
registration
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
cluster non empty basis of a2;
end;
:: YELLOW13:attrnot 1 => YELLOW13:attr 1
definition
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
let a3 be basis of a2;
attr a3 is correct means
for b1 being Element of bool the carrier of a1 holds
b1 in a3
iff
a2 in Int b1;
end;
:: YELLOW13:dfs 3
definiens
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
let a3 be basis of a2;
To prove
a3 is correct
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1 holds
b1 in a3
iff
a2 in Int b1;
:: YELLOW13:def 3
theorem
for b1 being TopStruct
for b2 being Element of the carrier of b1
for b3 being basis of b2 holds
b3 is correct(b1, b2)
iff
for b4 being Element of bool the carrier of b1 holds
b4 in b3
iff
b2 in Int b4;
:: YELLOW13:exreg 5
registration
let a1 be TopStruct;
let a2 be Element of the carrier of a1;
cluster correct basis of a2;
end;
:: YELLOW13:th 20
theorem
for b1 being TopStruct
for b2 being Element of the carrier of b1 holds
{b3 where b3 is Element of bool the carrier of b1: b2 in Int b3} is correct basis of b2;
:: YELLOW13:exreg 6
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster non empty correct basis of a2;
end;
:: YELLOW13:th 21
theorem
for b1 being non empty anti-discrete TopStruct
for b2 being Element of the carrier of b1 holds
{the carrier of b1} is correct basis of b2;
:: YELLOW13:th 22
theorem
for b1 being non empty anti-discrete TopStruct
for b2 being Element of the carrier of b1
for b3 being correct basis of b2 holds
b3 = {the carrier of b1};
:: YELLOW13:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2 holds
b3 is basis of b2;
:: YELLOW13:modenot 3 => YELLOW13:mode 2
definition
let a1 be TopStruct;
mode basis of A1 -> Element of bool bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
it is basis of b1;
end;
:: YELLOW13:dfs 4
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is basis of a1
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a2 is basis of b1;
:: YELLOW13:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is basis of b1
iff
for b3 being Element of the carrier of b1 holds
b2 is basis of b3;
:: YELLOW13:th 24
theorem
for b1 being TopStruct holds
bool the carrier of b1 is basis of b1;
:: YELLOW13:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being basis of b1 holds
b2 is not empty;
:: YELLOW13:condreg 10
registration
let a1 be non empty TopSpace-like TopStruct;
cluster -> non empty (basis of a1);
end;
:: YELLOW13:exreg 7
registration
let a1 be TopStruct;
cluster non empty basis of a1;
end;
:: YELLOW13:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being basis of b1 holds
the topology of b1 c= UniCl Int b2;
:: YELLOW13:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Basis of b1 holds
b2 is basis of b1;
:: YELLOW13:attrnot 2 => YELLOW13:attr 2
definition
let a1 be non empty TopSpace-like TopRelStr;
attr a1 is topological_semilattice means
for b1 being Function-like quasi_total Relation of the carrier of [:a1,a1:],the carrier of a1
st b1 = inf_op a1
holds b1 is continuous([:a1,a1:], a1);
end;
:: YELLOW13:dfs 5
definiens
let a1 be non empty TopSpace-like TopRelStr;
To prove
a1 is topological_semilattice
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of the carrier of [:a1,a1:],the carrier of a1
st b1 = inf_op a1
holds b1 is continuous([:a1,a1:], a1);
:: YELLOW13:def 5
theorem
for b1 being non empty TopSpace-like TopRelStr holds
b1 is topological_semilattice
iff
for b2 being Function-like quasi_total Relation of the carrier of [:b1,b1:],the carrier of b1
st b2 = inf_op b1
holds b2 is continuous([:b1,b1:], b1);
:: YELLOW13:condreg 11
registration
cluster non empty trivial TopSpace-like reflexive -> topological_semilattice (TopRelStr);
end;
:: YELLOW13:exreg 8
registration
cluster non empty trivial TopSpace-like reflexive TopRelStr;
end;
:: YELLOW13:th 28
theorem
for b1 being non empty TopSpace-like topological_semilattice TopRelStr
for b2 being Element of the carrier of b1 holds
b2 "/\" is continuous(b1, b1);
:: YELLOW13:funcreg 5
registration
let a1 be non empty TopSpace-like topological_semilattice TopRelStr;
let a2 be Element of the carrier of a1;
cluster a2 "/\" -> Function-like quasi_total continuous;
end;