Indicwiki: 1 revision imported from :alpha:Template:Families_of_sets

2023-03-07T08:43:08Z

1 revision imported from alpha:Template:Families_of_sets

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alpha>Indicwiki: Created page with "{| class="wikitable collapsible {{{floatposition|floatright}}} {{#ifeq:{{{1|}}}|expanded | |collapsed}}" style="{{#switch:{{{position|}}} |center|centre=margin-le..."

2023-02-16T19:59:34Z

Created page with "<includeonly>{| class="wikitable collapsible {{{floatposition|floatright}}} {{#ifeq:{{{1|}}}|expanded | |collapsed}}" style="{{#switch:{{{position|}}} |center|centre=margin-le..."

New page

<includeonly>{| class="wikitable collapsible {{{floatposition|floatright}}} {{#ifeq:{{{1|}}}|expanded | |collapsed}}" style="{{#switch:{{{position|}}} |center|centre=margin-left:auto;margin-right:auto; |#default=float:{{{position|right}}};}} margin-top:0; padding-top:0; text-align:center;"

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! colspan=11 style="text-align:center;padding-left:0.2em;padding-right:0.2em;font-size:90%;" | {{big|[[Family of sets|Families <math>\mathcal{F}</math> of sets]] over <math>\Omega</math>}} {{Navbar|Families of sets|mini=y|brackets=y|style=float:right; padding-left: 15px; padding-right: 5px;}}
|-
! style="padding:0.1;" | {{nowrap|{{small|Is necessarily true of <math>\mathcal{F}\colon</math>}}}} {{nowrap|{{small|or, is <math>\mathcal{F}</math> [[Closure (mathematics)|closed under:]]}}}}
| style="padding:0.1;" | {{small|[[Directed set|Directed by <math>\,\supseteq</math>]]}}
| style="padding:0.1;" | {{small|<math>A \cap B</math>}}
| style="padding:0.1;" | {{small|<math>A \cup B</math>}}
| style="padding:0.1;" | {{small|<math>B \setminus A</math>}}
| style="padding:0.1;" | {{small|<math>\Omega \setminus A</math>}}
| style="padding:0.1;" | {{small|<math>A_1 \cap A_2 \cap \cdots</math>}}
| style="padding:0.1;" | {{small|<math>A_1 \cup A_2 \cup \cdots</math>}}
| style="padding:0.1;" | {{small|<math>\Omega \in \mathcal{F}</math>}}
| style="padding:0.1;" | {{small|<math>\varnothing \in \mathcal{F}</math>}}
| style="padding:0.1;" | {{small|[[Finite intersection property|F.I.P.]]}}


|-
! {{nowrap|[[pi-system|{{pi}}-system]]}}
| {{ya}} || {{ya}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}}


|-
! {{nowrap|[[Ring of sets#semiring|Semiring]]}}
| {{ya}} || {{ya}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! {{nowrap|[[Semialgebra|{{nowrap|Semialgebra}} {{small|(Semifield)}}]]}}
| {{ya}} || {{ya}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[Monotone class|{{nowrap|Monotone class}}]]
| {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{ya|text=only if <math>A_i \searrow</math>}} || {{ya|text=only if <math>A_i \nearrow</math>}} || {{na}} || {{na}} || {{na}}


|-
! [[Dynkin system|{{nowrap|{{lambda}}-system}} {{nowrap|{{small|(Dynkin System)}}}}]]
| {{ya}} || {{na}} || {{na}} || {{ya|text=only if <math>A \subseteq B</math>}} || {{ya}} || {{na}} || {{ya|text=only if <math>A_i \nearrow</math> or they are [[Disjoint sets|disjoint]]}} || {{ya}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[Ring of sets|{{nowrap|Ring {{small|(Order theory)}}}}]]
| {{ya}} || {{ya}} || {{ya}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}} || {{na}}


|-
! [[Ring of sets|{{nowrap|Ring {{small|(Measure theory)}}}}]]
| {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{na}} || {{na}} || {{na}} || {{na}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[Delta-ring|{{nowrap|δ-Ring}}]]
| {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{na}} || {{ya}} || {{na}} || {{na}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[Sigma-ring|{{nowrap|{{sigma}}-Ring}}]]
| {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{na}} || {{ya}} || {{ya}} || {{na}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[Field of sets|{{nowrap|Algebra {{small|(Field)}}}}]]
| {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{na}} || {{na}} || {{ya}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[σ-algebra|{{nowrap|{{sigma}}-Algebra}} {{small|({{sigma}}-Field)}}]]
| {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[Dual ideal|{{nowrap|Dual ideal}}]]
| {{ya}} || {{ya}} || {{ya}} || {{na}} || {{na}} || {{na}} || {{ya}} || {{ya}} || {{na}} || {{na}}

|-
! [[Filter (set theory)|{{nowrap|Filter}}]]
| {{ya}} || {{ya}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never || {{na}} || {{ya}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|<math>\varnothing \not\in \mathcal{F}</math> || {{ya}}


|-
! [[Prefilter|{{nowrap|Prefilter}} {{small|(Filter base)}}]]
| {{ya}} || {{na}} || {{na}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never || {{na}} || {{na}} || {{na}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|<math>\varnothing \not\in \mathcal{F}</math> || {{ya}}


|-
! [[Filter subbase|{{nowrap|Filter subbase}}]]
| {{na}} || {{na}} || {{na}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never || {{na}} || {{na}} || {{na}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|<math>\varnothing \not\in \mathcal{F}</math> || {{ya}}


|-
! [[Topology (structure)|{{nowrap|Open Topology}}]]
| {{ya}} || {{ya}}|| {{ya}} || {{na}} || {{na}} || {{na}} || {{ya|text=[[File:Green check.svg|13px]] {{small|(even arbitrary <math>\cup</math>)}}}} || {{ya}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! [[Topology (structure)|{{nowrap|Closed Topology}}]]
| {{ya}} || {{ya}}|| {{ya}} || {{na}} || {{na}} || {{ya|text=[[File:Green check.svg|13px]] {{small|(even arbitrary <math>\cap</math>)}}}} || {{na}} || {{ya}} || {{ya}} || data-sort-value="No" style="background: #FF5252; color: black; vertical-align: middle; text-align: center;" class="table-no2"|Never


|-
! style="padding:0.1;" | {{nowrap|{{small|Is necessarily true of <math>\mathcal{F}\colon</math>}}}} {{nowrap|{{small|or, is <math>\mathcal{F}</math> [[Closure (mathematics)|closed under:]]}}}}
| style="padding:0.1;" | {{small|[[Directed downward|directed downward]]}}
| style="padding:0.1;" | {{small|{{nowrap|finite intersections}}}}
| style="padding:0.1;" | {{small|{{nowrap|finite unions}}}}
| style="padding:0.1;" | {{small|{{nowrap|relative complements}}}}
| style="padding:0.1;" | {{small|{{nowrap|complements in <math>\Omega</math>}}}}
| style="padding:0.1;" | {{small|{{nowrap|countable intersections}}}}
| style="padding:0.1;" | {{small|{{nowrap|countable unions}}}}
| style="padding:0.1;" | {{small|{{nowrap|contains <math>\Omega</math>}}}}
| style="padding:0.1;" | {{small|{{nowrap|contains <math>\varnothing</math>}}}}
| style="padding:0.1;" | {{small|[[Finite intersection property|Finite Intersection Property]]}}

|-
| colspan=11 style="text-align:center;" |
Additionally, a '''{{em|[[Ring of sets#semiring|semiring]]}}''' is a [[pi-system|{{pi}}-system]] where every complement <math>B \setminus A</math> is equal to a finite [[disjoint union]] of sets in <math>\mathcal{F}.</math> 
A '''{{em|[[semialgebra]]}}''' is a semiring that contains <math>\Omega.</math> 
<math>A, B, A_1, A_2, \ldots</math> are arbitrary elements of <math>\mathcal{F}</math> and it is assumed that <math>\mathcal{F} \neq \varnothing.</math> 

|}</includeonly><noinclude>{{clear}}{{doc}}
[[Category:Mathematics sidebar templates]]</noinclude>

Template:Families of sets - Revision history

Indicwiki: 1 revision imported from :alpha:Template:Families_of_sets

alpha>Indicwiki: Created page with "{| class="wikitable collapsible {{{floatposition|floatright}}} {{#ifeq:{{{1|}}}|expanded | |collapsed}}" style="{{#switch:{{{position|}}} |center|centre=margin-le..."