Article YELLOW_8, MML version 4.99.1005
:: YELLOW_8:th 1
theorem
for b1, b2, b3 being set
st b2 in Fin b1 & b3 c= b2
holds b3 in Fin b1;
:: YELLOW_8:th 2
theorem
for b1 being set
for b2 being Element of bool bool b1
st b2 c= Fin b1
holds meet b2 in Fin b1;
:: YELLOW_8:attrnot 1 => REALSET1:attr 1
definition
let a1 be set;
attr a1 is trivial means
for b1, b2 being Element of a1 holds
b1 = b2;
end;
:: YELLOW_8:dfs 1
definiens
let a1 be non empty set;
To prove
a1 is trivial
it is sufficient to prove
thus for b1, b2 being Element of a1 holds
b1 = b2;
:: YELLOW_8:def 1
theorem
for b1 being non empty set holds
b1 is trivial
iff
for b2, b3 being Element of b1 holds
b2 = b3;
:: YELLOW_8:th 4
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2,COMPLEMENT b2 are_equipotent;
:: YELLOW_8:th 5
theorem
for b1, b2 being set
st b1,b2 are_equipotent & b1 is countable
holds b2 is countable;
:: YELLOW_8:th 14
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 ` in COMPLEMENT b2
iff
b3 in b2;
:: YELLOW_8:th 15
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
Intersect COMPLEMENT b2 = (union b2) `;
:: YELLOW_8:th 16
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
union COMPLEMENT b2 = (Intersect b2) `;
:: YELLOW_8:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b3 c= b2 &
b2 is closed(b1) &
(for b4 being Element of bool the carrier of b1
st b3 c= b4 & b4 is closed(b1)
holds b2 c= b4)
holds b2 = Cl b3;
:: YELLOW_8:th 18
theorem
for b1 being TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1
st b3 is open(b1)
holds b3 = union {b4 where b4 is Element of bool the carrier of b1: b4 in b2 & b4 c= b3};
:: YELLOW_8:th 19
theorem
for b1 being TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is open(b1);
:: YELLOW_8:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1 holds
Int b3 = union {b4 where b4 is Element of bool the carrier of b1: b4 in b2 & b4 c= b3};
:: YELLOW_8:modenot 1 => YELLOW_8:mode 1
definition
let a1 be non empty TopStruct;
let a2 be Element of the carrier of a1;
mode Basis of A2 -> Element of bool bool the carrier of a1 means
it c= the topology of a1 &
a2 in Intersect it &
(for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a2 in b1
holds ex b2 being Element of bool the carrier of a1 st
b2 in it & b2 c= b1);
end;
:: YELLOW_8:dfs 2
definiens
let a1 be non empty 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 a3 c= the topology of a1 &
a2 in Intersect a3 &
(for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a2 in b1
holds ex b2 being Element of bool the carrier of a1 st
b2 in a3 & b2 c= b1);
:: YELLOW_8:def 2
theorem
for b1 being non empty 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
b3 c= the topology of b1 &
b2 in Intersect b3 &
(for b4 being Element of bool the carrier of b1
st b4 is open(b1) & b2 in b4
holds ex b5 being Element of bool the carrier of b1 st
b5 in b3 & b5 c= b4);
:: YELLOW_8:th 21
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2
for b4 being Element of bool the carrier of b1
st b4 in b3
holds b4 is open(b1) & b2 in b4;
:: YELLOW_8:th 22
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2
for b4 being Element of bool the carrier of b1
st b2 in b4 & b4 is open(b1)
holds ex b5 being Element of bool the carrier of b1 st
b5 in b3 & b5 c= b4;
:: YELLOW_8:th 23
theorem
for b1 being non empty TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 c= the topology of b1 &
(for b3 being Element of the carrier of b1 holds
ex b4 being Basis of b3 st
b4 c= b2)
holds b2 is Basis of b1;
:: YELLOW_8:attrnot 2 => 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 everywhere_dense(a1))
holds ex b2 being Element of bool the carrier of a1 st
b2 = Intersect b1 & b2 is dense(a1);
end;
:: YELLOW_8:dfs 3
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 everywhere_dense(a1))
holds ex b2 being Element of bool the carrier of a1 st
b2 = Intersect b1 & b2 is dense(a1);
:: YELLOW_8:def 3
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 everywhere_dense(b1))
holds ex b3 being Element of bool the carrier of b1 st
b3 = Intersect b2 & b3 is dense(b1);
:: YELLOW_8:th 24
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 nowhere_dense(b1))
holds union b2 is boundary(b1);
:: YELLOW_8:attrnot 3 => YELLOW_8:attr 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is irreducible means
a2 is not empty &
a2 is closed(a1) &
(for b1, b2 being Element of bool the carrier of a1
st b1 is closed(a1) & b2 is closed(a1) & a2 = b1 \/ b2 & b1 <> a2
holds b2 = a2);
end;
:: YELLOW_8:dfs 4
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is irreducible
it is sufficient to prove
thus a2 is not empty &
a2 is closed(a1) &
(for b1, b2 being Element of bool the carrier of a1
st b1 is closed(a1) & b2 is closed(a1) & a2 = b1 \/ b2 & b1 <> a2
holds b2 = a2);
:: YELLOW_8:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is irreducible(b1)
iff
b2 is not empty &
b2 is closed(b1) &
(for b3, b4 being Element of bool the carrier of b1
st b3 is closed(b1) & b4 is closed(b1) & b2 = b3 \/ b4 & b3 <> b2
holds b4 = b2);
:: YELLOW_8:condreg 1
registration
let a1 be TopStruct;
cluster irreducible -> non empty (Element of bool the carrier of a1);
end;
:: YELLOW_8:prednot 1 => YELLOW_8:pred 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
pred A3 is_dense_point_of A2 means
a3 in a2 & a2 c= Cl {a3};
end;
:: YELLOW_8:dfs 5
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
To prove
a3 is_dense_point_of a2
it is sufficient to prove
thus a3 in a2 & a2 c= Cl {a3};
:: YELLOW_8:def 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 is_dense_point_of b2
iff
b3 in b2 & b2 c= Cl {b3};
:: YELLOW_8:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
for b3 being Element of the carrier of b1
st b3 is_dense_point_of b2
holds b2 = Cl {b3};
:: YELLOW_8:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
Cl {b2} is irreducible(b1);
:: YELLOW_8:exreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
cluster irreducible Element of bool the carrier of a1;
end;
:: YELLOW_8:attrnot 4 => YELLOW_8:attr 3
definition
let a1 be non empty TopSpace-like TopStruct;
attr a1 is sober means
for b1 being irreducible Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 is_dense_point_of b1 &
(for b3 being Element of the carrier of a1
st b3 is_dense_point_of b1
holds b2 = b3);
end;
:: YELLOW_8:dfs 6
definiens
let a1 be non empty TopSpace-like TopStruct;
To prove
a1 is sober
it is sufficient to prove
thus for b1 being irreducible Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 is_dense_point_of b1 &
(for b3 being Element of the carrier of a1
st b3 is_dense_point_of b1
holds b2 = b3);
:: YELLOW_8:def 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is sober
iff
for b2 being irreducible Element of bool the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b3 is_dense_point_of b2 &
(for b4 being Element of the carrier of b1
st b4 is_dense_point_of b2
holds b3 = b4);
:: YELLOW_8:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
b2 is_dense_point_of Cl {b2};
:: YELLOW_8:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
b2 is_dense_point_of {b2};
:: YELLOW_8:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1) & b3 is closed(b1)
holds b3 \ b2 is closed(b1);
:: YELLOW_8:th 30
theorem
for b1 being non empty TopSpace-like being_T2 TopStruct
for b2 being irreducible Element of bool the carrier of b1 holds
b2 is trivial;
:: YELLOW_8:condreg 2
registration
let a1 be non empty TopSpace-like being_T2 TopStruct;
cluster irreducible -> trivial (Element of bool the carrier of a1);
end;
:: YELLOW_8:th 31
theorem
for b1 being non empty TopSpace-like being_T2 TopStruct holds
b1 is sober;
:: YELLOW_8:condreg 3
registration
cluster non empty TopSpace-like being_T2 -> sober (TopStruct);
end;
:: YELLOW_8:exreg 2
registration
cluster non empty TopSpace-like sober TopStruct;
end;
:: YELLOW_8:th 32
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is discerning
iff
for b2, b3 being Element of the carrier of b1
st Cl {b2} = Cl {b3}
holds b2 = b3;
:: YELLOW_8:th 33
theorem
for b1 being non empty TopSpace-like sober TopStruct holds
b1 is discerning;
:: YELLOW_8:condreg 4
registration
cluster non empty TopSpace-like sober -> discerning (TopStruct);
end;
:: YELLOW_8:funcnot 1 => YELLOW_8:func 1
definition
let a1 be set;
func CofinTop A1 -> strict TopStruct means
the carrier of it = a1 &
COMPLEMENT the topology of it = {a1} \/ Fin a1;
end;
:: YELLOW_8:def 7
theorem
for b1 being set
for b2 being strict TopStruct holds
b2 = CofinTop b1
iff
the carrier of b2 = b1 &
COMPLEMENT the topology of b2 = {b1} \/ Fin b1;
:: YELLOW_8:funcreg 1
registration
let a1 be non empty set;
cluster CofinTop a1 -> non empty strict;
end;
:: YELLOW_8:funcreg 2
registration
let a1 be set;
cluster CofinTop a1 -> strict TopSpace-like;
end;
:: YELLOW_8:th 34
theorem
for b1 being non empty set
for b2 being Element of bool the carrier of CofinTop b1 holds
b2 is closed(CofinTop b1)
iff
(b2 = b1 or b2 is finite);
:: YELLOW_8:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T1
for b2 being Element of the carrier of b1 holds
Cl {b2} = {b2};
:: YELLOW_8:funcreg 3
registration
let a1 be non empty set;
cluster CofinTop a1 -> strict being_T1;
end;
:: YELLOW_8:funcreg 4
registration
let a1 be infinite set;
cluster CofinTop a1 -> strict non sober;
end;
:: YELLOW_8:exreg 3
registration
cluster non empty TopSpace-like being_T1 non sober TopStruct;
end;
:: YELLOW_8:th 36
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is being_T3
iff
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b2 in Int b3
holds ex b4 being Element of bool the carrier of b1 st
b4 is closed(b1) & b4 c= b3 & b2 in Int b4;
:: YELLOW_8:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T3
holds b1 is locally-compact
iff
for b2 being Element of the carrier of b1 holds
ex b3 being Element of bool the carrier of b1 st
b2 in Int b3 & b3 is compact(b1);