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Template's default state when transcluded is collapsed. To override, invoke as {{Families of sets|expanded}}
.
To change the template's position from the default shown, add the parameter position
with the value "left
", "center
", "centre
" or "right
".
Calling
{{ Families of sets }}
will display:
Families
F
{\displaystyle {\mathcal {F}}}
of sets over
Ω
{\displaystyle \Omega }
Is necessarily true of
F
:
{\displaystyle {\mathcal {F}}\colon }
or, is
F
{\displaystyle {\mathcal {F}}}
closed under:
Directed by
⊇
{\displaystyle \,\supseteq }
A
∩
B
{\displaystyle A\cap B}
A
∪
B
{\displaystyle A\cup B}
B
∖
A
{\displaystyle B\setminus A}
Ω
∖
A
{\displaystyle \Omega \setminus A}
A
1
∩
A
2
∩
⋯
{\displaystyle A_{1}\cap A_{2}\cap \cdots }
A
1
∪
A
2
∪
⋯
{\displaystyle A_{1}\cup A_{2}\cup \cdots }
Ω
∈
F
{\displaystyle \Omega \in {\mathcal {F}}}
∅
∈
F
{\displaystyle \varnothing \in {\mathcal {F}}}
F.I.P.
π -system
Semiring
Never
Semialgebra (Semifield)
Never
Monotone class
only if
A
i
↘
{\displaystyle A_{i}\searrow }
only if
A
i
↗
{\displaystyle A_{i}\nearrow }
𝜆-system (Dynkin System)
only if
A
⊆
B
{\displaystyle A\subseteq B}
only if
A
i
↗
{\displaystyle A_{i}\nearrow }
or they are disjoint
Never
Ring (Order theory)
Ring (Measure theory)
Never
δ-Ring
Never
𝜎-Ring
Never
Algebra (Field)
Never
𝜎-Algebra (𝜎-Field)
Never
Dual ideal
Filter
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Prefilter (Filter base)
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Filter subbase
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Open Topology
(even arbitrary
∪
{\displaystyle \cup }
)
Never
Closed Topology
(even arbitrary
∩
{\displaystyle \cap }
)
Never
Is necessarily true of
F
:
{\displaystyle {\mathcal {F}}\colon }
or, is
F
{\displaystyle {\mathcal {F}}}
closed under:
directed downward
finite intersections
finite unions
relative complements
complements in
Ω
{\displaystyle \Omega }
countable intersections
countable unions
contains
Ω
{\displaystyle \Omega }
contains
∅
{\displaystyle \varnothing }
Finite Intersection Property
Additionally, a semiring is a π -system where every complement
B
∖
A
{\displaystyle B\setminus A}
is equal to a finite disjoint union of sets in
F
.
{\displaystyle {\mathcal {F}}.}
A semialgebra is a semiring where every complement
Ω
∖
A
{\displaystyle \Omega \setminus A}
is equal to a finite disjoint union of sets in
F
.
{\displaystyle {\mathcal {F}}.}
A
,
B
,
A
1
,
A
2
,
…
{\displaystyle A,B,A_{1},A_{2},\ldots }
are arbitrary elements of
F
{\displaystyle {\mathcal {F}}}
and it is assumed that
F
≠
∅
.
{\displaystyle {\mathcal {F}}\neq \varnothing .}
Call with alignment [ edit ]
Calling
{{Families of sets|position=left}}
will display:
Families
F
{\displaystyle {\mathcal {F}}}
of sets over
Ω
{\displaystyle \Omega }
Is necessarily true of
F
:
{\displaystyle {\mathcal {F}}\colon }
or, is
F
{\displaystyle {\mathcal {F}}}
closed under:
Directed by
⊇
{\displaystyle \,\supseteq }
A
∩
B
{\displaystyle A\cap B}
A
∪
B
{\displaystyle A\cup B}
B
∖
A
{\displaystyle B\setminus A}
Ω
∖
A
{\displaystyle \Omega \setminus A}
A
1
∩
A
2
∩
⋯
{\displaystyle A_{1}\cap A_{2}\cap \cdots }
A
1
∪
A
2
∪
⋯
{\displaystyle A_{1}\cup A_{2}\cup \cdots }
Ω
∈
F
{\displaystyle \Omega \in {\mathcal {F}}}
∅
∈
F
{\displaystyle \varnothing \in {\mathcal {F}}}
F.I.P.
π -system
Semiring
Never
Semialgebra (Semifield)
Never
Monotone class
only if
A
i
↘
{\displaystyle A_{i}\searrow }
only if
A
i
↗
{\displaystyle A_{i}\nearrow }
𝜆-system (Dynkin System)
only if
A
⊆
B
{\displaystyle A\subseteq B}
only if
A
i
↗
{\displaystyle A_{i}\nearrow }
or they are disjoint
Never
Ring (Order theory)
Ring (Measure theory)
Never
δ-Ring
Never
𝜎-Ring
Never
Algebra (Field)
Never
𝜎-Algebra (𝜎-Field)
Never
Dual ideal
Filter
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Prefilter (Filter base)
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Filter subbase
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Open Topology
(even arbitrary
∪
{\displaystyle \cup }
)
Never
Closed Topology
(even arbitrary
∩
{\displaystyle \cap }
)
Never
Is necessarily true of
F
:
{\displaystyle {\mathcal {F}}\colon }
or, is
F
{\displaystyle {\mathcal {F}}}
closed under:
directed downward
finite intersections
finite unions
relative complements
complements in
Ω
{\displaystyle \Omega }
countable intersections
countable unions
contains
Ω
{\displaystyle \Omega }
contains
∅
{\displaystyle \varnothing }
Finite Intersection Property
Additionally, a semiring is a π -system where every complement
B
∖
A
{\displaystyle B\setminus A}
is equal to a finite disjoint union of sets in
F
.
{\displaystyle {\mathcal {F}}.}
A semialgebra is a semiring where every complement
Ω
∖
A
{\displaystyle \Omega \setminus A}
is equal to a finite disjoint union of sets in
F
.
{\displaystyle {\mathcal {F}}.}
A
,
B
,
A
1
,
A
2
,
…
{\displaystyle A,B,A_{1},A_{2},\ldots }
are arbitrary elements of
F
{\displaystyle {\mathcal {F}}}
and it is assumed that
F
≠
∅
.
{\displaystyle {\mathcal {F}}\neq \varnothing .}
Expanded with alignment [ edit ]
Calling
{{Families of sets|expanded|position=left}}
will display:
Families
F
{\displaystyle {\mathcal {F}}}
of sets over
Ω
{\displaystyle \Omega }
Is necessarily true of
F
:
{\displaystyle {\mathcal {F}}\colon }
or, is
F
{\displaystyle {\mathcal {F}}}
closed under:
Directed by
⊇
{\displaystyle \,\supseteq }
A
∩
B
{\displaystyle A\cap B}
A
∪
B
{\displaystyle A\cup B}
B
∖
A
{\displaystyle B\setminus A}
Ω
∖
A
{\displaystyle \Omega \setminus A}
A
1
∩
A
2
∩
⋯
{\displaystyle A_{1}\cap A_{2}\cap \cdots }
A
1
∪
A
2
∪
⋯
{\displaystyle A_{1}\cup A_{2}\cup \cdots }
Ω
∈
F
{\displaystyle \Omega \in {\mathcal {F}}}
∅
∈
F
{\displaystyle \varnothing \in {\mathcal {F}}}
F.I.P.
π -system
Semiring
Never
Semialgebra (Semifield)
Never
Monotone class
only if
A
i
↘
{\displaystyle A_{i}\searrow }
only if
A
i
↗
{\displaystyle A_{i}\nearrow }
𝜆-system (Dynkin System)
only if
A
⊆
B
{\displaystyle A\subseteq B}
only if
A
i
↗
{\displaystyle A_{i}\nearrow }
or they are disjoint
Never
Ring (Order theory)
Ring (Measure theory)
Never
δ-Ring
Never
𝜎-Ring
Never
Algebra (Field)
Never
𝜎-Algebra (𝜎-Field)
Never
Dual ideal
Filter
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Prefilter (Filter base)
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Filter subbase
Never
Never
∅
∉
F
{\displaystyle \varnothing \not \in {\mathcal {F}}}
Open Topology
(even arbitrary
∪
{\displaystyle \cup }
)
Never
Closed Topology
(even arbitrary
∩
{\displaystyle \cap }
)
Never
Is necessarily true of
F
:
{\displaystyle {\mathcal {F}}\colon }
or, is
F
{\displaystyle {\mathcal {F}}}
closed under:
directed downward
finite intersections
finite unions
relative complements
complements in
Ω
{\displaystyle \Omega }
countable intersections
countable unions
contains
Ω
{\displaystyle \Omega }
contains
∅
{\displaystyle \varnothing }
Finite Intersection Property
Additionally, a semiring is a π -system where every complement
B
∖
A
{\displaystyle B\setminus A}
is equal to a finite disjoint union of sets in
F
.
{\displaystyle {\mathcal {F}}.}
A semialgebra is a semiring where every complement
Ω
∖
A
{\displaystyle \Omega \setminus A}
is equal to a finite disjoint union of sets in
F
.
{\displaystyle {\mathcal {F}}.}
A
,
B
,
A
1
,
A
2
,
…
{\displaystyle A,B,A_{1},A_{2},\ldots }
are arbitrary elements of
F
{\displaystyle {\mathcal {F}}}
and it is assumed that
F
≠
∅
.
{\displaystyle {\mathcal {F}}\neq \varnothing .}