Article WAYBEL12, MML version 4.99.1005
:: WAYBEL12:attrnot 1 => PRE_TOPC:attr 4
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is closed means
a2 ` is open(a1);
end;
:: WAYBEL12:dfs 1
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is closed
it is sufficient to prove
thus a2 ` is open(a1);
:: WAYBEL12:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
b2 ` is open(b1);
:: WAYBEL12:attrnot 2 => WAYBEL12:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is dense means
for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is dense(a1);
end;
:: WAYBEL12:dfs 2
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is dense
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is dense(a1);
:: WAYBEL12:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is dense(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is dense(b1);
:: WAYBEL12:condreg 1
registration
let a1 be empty 1-sorted;
cluster -> empty (Element of bool the carrier of a1);
end;
:: WAYBEL12:condreg 2
registration
cluster finite -> countable (set);
end;
:: WAYBEL12:exreg 1
registration
cluster non empty finite set;
end;
:: WAYBEL12:exreg 2
registration
let a1 be 1-sorted;
cluster empty Element of bool the carrier of a1;
end;
:: WAYBEL12:exreg 3
registration
let a1 be non empty RelStr;
cluster non empty finite Element of bool the carrier of a1;
end;
:: WAYBEL12:exreg 4
registration
cluster countable infinite set;
end;
:: WAYBEL12:exreg 5
registration
let a1 be 1-sorted;
cluster empty Element of bool bool the carrier of a1;
end;
:: WAYBEL12:th 2
theorem
for b1, b2 being set
st Card b1 c= Card b2 & b2 is countable
holds b1 is countable;
:: WAYBEL12:th 3
theorem
for b1 being countable infinite set holds
NAT,b1 are_equipotent;
:: WAYBEL12:th 4
theorem
for b1 being non empty countable set holds
ex b2 being Function-like quasi_total Relation of NAT,b1 st
rng b2 = b1;
:: WAYBEL12:th 7
theorem
for b1 being non empty transitive RelStr
for b2, b3 being Element of bool the carrier of b1
st b2 is_finer_than b3
holds downarrow b2 c= downarrow b3;
:: WAYBEL12:th 8
theorem
for b1 being non empty transitive RelStr
for b2, b3 being Element of bool the carrier of b1
st b2 is_coarser_than b3
holds uparrow b2 c= uparrow b3;
:: WAYBEL12:th 9
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty finite filtered Element of bool the carrier of b1
st ex_inf_of b2,b1
holds "/\"(b2,b1) in b2;
:: WAYBEL12:th 10
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being non empty lower Element of bool the carrier of b1 holds
Bottom b1 in b2;
:: WAYBEL12:th 11
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being non empty Element of bool the carrier of b1 holds
Bottom b1 in downarrow b2;
:: WAYBEL12:th 12
theorem
for b1 being non empty antisymmetric upper-bounded RelStr
for b2 being non empty upper Element of bool the carrier of b1 holds
Top b1 in b2;
:: WAYBEL12:th 13
theorem
for b1 being non empty antisymmetric upper-bounded RelStr
for b2 being non empty Element of bool the carrier of b1 holds
Top b1 in uparrow b2;
:: WAYBEL12:th 14
theorem
for b1 being antisymmetric lower-bounded with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
b2 "/\" {Bottom b1} c= {Bottom b1};
:: WAYBEL12:th 15
theorem
for b1 being antisymmetric lower-bounded with_infima RelStr
for b2 being non empty Element of bool the carrier of b1 holds
b2 "/\" {Bottom b1} = {Bottom b1};
:: WAYBEL12:th 16
theorem
for b1 being antisymmetric upper-bounded with_suprema RelStr
for b2 being Element of bool the carrier of b1 holds
b2 "\/" {Top b1} c= {Top b1};
:: WAYBEL12:th 17
theorem
for b1 being antisymmetric upper-bounded with_suprema RelStr
for b2 being non empty Element of bool the carrier of b1 holds
b2 "\/" {Top b1} = {Top b1};
:: WAYBEL12:th 18
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
{Top b1} "/\" b2 = b2;
:: WAYBEL12:th 19
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema RelStr
for b2 being Element of bool the carrier of b1 holds
{Bottom b1} "\/" b2 = b2;
:: WAYBEL12:th 20
theorem
for b1 being non empty reflexive RelStr
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds b2 is_finer_than b3 & b2 is_coarser_than b3;
:: WAYBEL12:th 21
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 <= b4
holds {b4} "/\" b2 is_coarser_than {b3} "/\" b2;
:: WAYBEL12:th 22
theorem
for b1 being transitive antisymmetric with_suprema RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 <= b4
holds {b3} "\/" b2 is_finer_than {b4} "\/" b2;
:: WAYBEL12:th 23
theorem
for b1 being non empty RelStr
for b2, b3, b4 being Element of bool the carrier of b1
st b3 is_coarser_than b4 & b2 is upper(b1) & b4 c= b2
holds b3 c= b2;
:: WAYBEL12:th 24
theorem
for b1 being non empty RelStr
for b2, b3, b4 being Element of bool the carrier of b1
st b3 is_finer_than b4 & b2 is lower(b1) & b4 c= b2
holds b3 c= b2;
:: WAYBEL12:th 25
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being filtered upper Element of bool the carrier of b1 holds
b2 "/\" b2 = b2;
:: WAYBEL12:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being directed lower Element of bool the carrier of b1 holds
b2 "\/" b2 = b2;
:: WAYBEL12:th 27
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
{b3 where b3 is Element of the carrier of b1: b2 "/\" {b3} c= b2} is not empty;
:: WAYBEL12:th 28
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2 being Element of bool the carrier of b1 holds
{b3 where b3 is Element of the carrier of b1: b2 "/\" {b3} c= b2} is filtered Element of bool the carrier of b1;
:: WAYBEL12:th 29
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2 being upper Element of bool the carrier of b1 holds
{b3 where b3 is Element of the carrier of b1: b2 "/\" {b3} c= b2} is upper Element of bool the carrier of b1;
:: WAYBEL12:th 30
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of bool the carrier of b1
st b2 is Open(b1) & b2 is lower(b1)
holds b2 is filtered(b1);
:: WAYBEL12:condreg 3
registration
let a1 be reflexive transitive antisymmetric with_infima RelStr;
cluster lower Open -> filtered (Element of bool the carrier of a1);
end;
:: WAYBEL12:condreg 4
registration
let a1 be non empty reflexive antisymmetric continuous RelStr;
cluster lower -> Open (Element of bool the carrier of a1);
end;
:: WAYBEL12:funcreg 1
registration
let a1 be reflexive transitive antisymmetric with_infima continuous RelStr;
let a2 be Element of the carrier of a1;
cluster (downarrow a2) ` -> Open;
end;
:: WAYBEL12:th 31
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty Element of bool the carrier of b1
st for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & not b3 <= b4
holds b4 <= b3
for b3 being non empty finite Element of bool b2 holds
"/\"(b3,b1) in b3;
:: WAYBEL12:th 32
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty Element of bool the carrier of b1
st for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & not b3 <= b4
holds b4 <= b3
for b3 being non empty finite Element of bool b2 holds
"\/"(b3,b1) in b3;
:: WAYBEL12:modenot 1 => WAYBEL12:mode 1
definition
let a1 be reflexive transitive antisymmetric with_infima RelStr;
let a2 be non empty filtered upper Element of bool the carrier of a1;
mode GeneratorSet of A2 -> Element of bool the carrier of a1 means
a2 = uparrow fininfs it;
end;
:: WAYBEL12:dfs 3
definiens
let a1 be reflexive transitive antisymmetric with_infima RelStr;
let a2 be non empty filtered upper Element of bool the carrier of a1;
let a3 be Element of bool the carrier of a1;
To prove
a3 is GeneratorSet of a2
it is sufficient to prove
thus a2 = uparrow fininfs a3;
:: WAYBEL12:def 3
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 is GeneratorSet of b2
iff
b2 = uparrow fininfs b3;
:: WAYBEL12:exreg 6
registration
let a1 be reflexive transitive antisymmetric with_infima RelStr;
let a2 be non empty filtered upper Element of bool the carrier of a1;
cluster non empty GeneratorSet of a2;
end;
:: WAYBEL12:th 33
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of bool the carrier of b1
for b3 being non empty Element of bool the carrier of b1
st b2 is_coarser_than b3
holds fininfs b2 is_coarser_than fininfs b3;
:: WAYBEL12:th 34
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1
for b3 being GeneratorSet of b2
for b4 being non empty Element of bool the carrier of b1
st b3 is_coarser_than b4 & b4 is_coarser_than b2
holds b4 is GeneratorSet of b2;
:: WAYBEL12:th 35
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 = b2 &
(for b5 being Element of NAT holds
b4 . b5 = "/\"({b3 . b6 where b6 is Element of NAT: b6 <= b5},b1))
holds b2 is_coarser_than rng b4;
:: WAYBEL12:th 36
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1
for b3 being GeneratorSet of b2
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b4 = b3 &
(for b6 being Element of NAT holds
b5 . b6 = "/\"({b4 . b7 where b7 is Element of NAT: b7 <= b6},b1))
holds rng b5 is GeneratorSet of b2;
:: WAYBEL12:th 37
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being upper Open Element of bool the carrier of b1
for b3 being non empty filtered upper Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b2 "/\" b3 c= b2 &
b4 in b2 &
(ex b5 being non empty GeneratorSet of b3 st
b5 is countable)
holds ex b5 being non empty filtered upper Open Element of bool the carrier of b1 st
b5 c= b2 & b4 in b5 & b3 c= b5;
:: WAYBEL12:th 38
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being upper Open Element of bool the carrier of b1
for b3 being non empty countable Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b2 "/\" b3 c= b2 & b4 in b2
holds ex b5 being non empty filtered upper Open Element of bool the carrier of b1 st
{b4} "/\" b3 c= b5 & b5 c= b2 & b4 in b5;
:: WAYBEL12:th 39
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being upper Open Element of bool the carrier of b1
for b3 being non empty countable Element of bool the carrier of b1
for b4, b5 being Element of the carrier of b1
st b2 "/\" b3 c= b2 & b5 in b2 & not b4 in b2
holds ex b6 being meet-irreducible Element of the carrier of b1 st
b4 <= b6 & not b6 in uparrow ({b5} "/\" b3);
:: WAYBEL12:th 40
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
for b3 being non empty countable Element of bool the carrier of b1
st for b4, b5 being Element of the carrier of b1
st not b5 <= b2 & b4 in b3
holds not b5 "/\" b4 <= b2
for b4 being Element of the carrier of b1
st not b4 <= b2
holds ex b5 being meet-irreducible Element of the carrier of b1 st
b2 <= b5 & not b5 in uparrow ({b4} "/\" b3);
:: WAYBEL12:attrnot 3 => WAYBEL12:attr 2
definition
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
attr a2 is dense means
for b1 being Element of the carrier of a1
st b1 <> Bottom a1
holds a2 "/\" b1 <> Bottom a1;
end;
:: WAYBEL12:dfs 4
definiens
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
To prove
a2 is dense
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st b1 <> Bottom a1
holds a2 "/\" b1 <> Bottom a1;
:: WAYBEL12:def 4
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1 holds
b2 is dense(b1)
iff
for b3 being Element of the carrier of b1
st b3 <> Bottom b1
holds b2 "/\" b3 <> Bottom b1;
:: WAYBEL12:funcreg 2
registration
let a1 be reflexive transitive antisymmetric upper-bounded with_infima RelStr;
cluster Top a1 -> dense;
end;
:: WAYBEL12:exreg 7
registration
let a1 be reflexive transitive antisymmetric upper-bounded with_infima RelStr;
cluster dense Element of the carrier of a1;
end;
:: WAYBEL12:th 41
theorem
for b1 being non trivial reflexive transitive antisymmetric bounded with_infima RelStr
for b2 being Element of the carrier of b1
st b2 is dense(b1)
holds b2 <> Bottom b1;
:: WAYBEL12:attrnot 4 => WAYBEL12:attr 3
definition
let a1 be non empty RelStr;
let a2 be Element of bool the carrier of a1;
attr a2 is dense means
for b1 being Element of the carrier of a1
st b1 in a2
holds b1 is dense(a1);
end;
:: WAYBEL12:dfs 5
definiens
let a1 be non empty RelStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is dense
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st b1 in a2
holds b1 is dense(a1);
:: WAYBEL12:def 5
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is dense(b1)
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds b3 is dense(b1);
:: WAYBEL12:th 42
theorem
for b1 being reflexive transitive antisymmetric upper-bounded with_infima RelStr holds
{Top b1} is dense(b1);
:: WAYBEL12:exreg 8
registration
let a1 be reflexive transitive antisymmetric upper-bounded with_infima RelStr;
cluster non empty countable finite dense Element of bool the carrier of a1;
end;
:: WAYBEL12:th 43
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
for b2 being non empty countable dense Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 <> Bottom b1
holds ex b4 being meet-irreducible Element of the carrier of b1 st
b4 <> Top b1 & not b4 in uparrow ({b3} "/\" b2);
:: WAYBEL12:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of InclPoset the topology of b1
for b3 being Element of bool the carrier of b1
st b2 = b3 & b3 ` is irreducible(b1)
holds b2 is meet-irreducible(InclPoset the topology of b1);
:: WAYBEL12:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of InclPoset the topology of b1
for b3 being Element of bool the carrier of b1
st b2 = b3 & b2 <> Top InclPoset the topology of b1
holds b2 is meet-irreducible(InclPoset the topology of b1)
iff
b3 ` is irreducible(b1);
:: WAYBEL12:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of InclPoset the topology of b1
for b3 being Element of bool the carrier of b1
st b2 = b3
holds b2 is dense(InclPoset the topology of b1)
iff
b3 is everywhere_dense(b1);
:: WAYBEL12:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is locally-compact
for b2 being countable Element of bool bool the carrier of b1
st b2 is not empty & b2 is dense(b1) & b2 is open(b1)
for b3 being non empty Element of bool the carrier of b1
st b3 is open(b1)
holds ex b4 being irreducible Element of bool the carrier of b1 st
for b5 being Element of bool the carrier of b1
st b5 in b2
holds b4 /\ b3 meets b5;
:: WAYBEL12:attrnot 5 => YELLOW_8:attr 1
definition
let a1 be non empty TopSpace-like TopStruct;
attr a1 is Baire means
for b1 being Element of bool bool the carrier of a1
st b1 is countable &
(for b2 being Element of bool the carrier of a1
st b2 in b1
holds b2 is open(a1) & b2 is dense(a1))
holds ex b2 being Element of bool the carrier of a1 st
b2 = Intersect b1 & b2 is dense(a1);
end;
:: WAYBEL12:dfs 6
definiens
let a1 be non empty TopSpace-like TopStruct;
To prove
a1 is Baire
it is sufficient to prove
thus for b1 being Element of bool bool the carrier of a1
st b1 is countable &
(for b2 being Element of bool the carrier of a1
st b2 in b1
holds b2 is open(a1) & b2 is dense(a1))
holds ex b2 being Element of bool the carrier of a1 st
b2 = Intersect b1 & b2 is dense(a1);
:: WAYBEL12:def 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is Baire
iff
for b2 being Element of bool bool the carrier of b1
st b2 is countable &
(for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is open(b1) & b3 is dense(b1))
holds ex b3 being Element of bool the carrier of b1 st
b3 = Intersect b2 & b3 is dense(b1);
:: WAYBEL12:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is sober & b1 is locally-compact
holds b1 is Baire;