Title: Theory / Unimportant / Math / Analysis / Non-standard / Goldblatt+15 / Filter
Tags: récupéré

Name: A filter of \(I\)
Definition: closed under finite intersection
Page: 18
Chapter: 2
Variables: \(I\) a set
Definition2: closed under taking superset
Conjdef: A class of subsets of \(I\)
Definition3: non-empty
Definition4: Equivalently, \(A\cap B\) belongs to this set iff \(A,B\) belong to this set
Denoted by: \(\mathcal F\)

Name: A proper filter of \(I\)
Definition: A filter of \(I\) not containing empty set
Page: 18
Chapter: 2
Variables: \(I\) a set

Name: An ultrafilter of \(I\)
Definition: for each \(A\subseteq\mathbb N\), either \(A\) or \(I\setminus A\) belong to this class
Page: 17, 19
Chapter: 2
Section: 2, 5
Variables: \(I\) a set
Index: , (6)
Definition2: \(\subseteq\)-maximal proper filter
Conjdef: A filter of \(I\) such that, equivalently

Name: Principal ultrafilter generated by \(i\)
Definition: the set of subsets of \(I\) containing \(i\)
Page: 18
Chapter: 2
Section: 4
Variables: \(I\) a set,
\(i\in I\)
Notation: \(\mathcal F^i\)
Index: (1)
Kind: Ex
Typ: ultrafilter

Name: Cofinite filter on \(I\)
Definition: The set of cofinite subsets of \(I\)
Page: 18
Chapter: 2
Section: 4
Variables: \(I\) a set,
Notation: \(\mathcal F^{co}\)
Index: (2)
Kind: Ex
Name2: Fréchet filter on \(I\)
Typ: Filter, not ultra

Name: Filter generated by \(\mathcal H\)
Definition: Smallest filter of \(I\) including \(\mathcal H\) in the collection
Page: 18
Chapter: 2
Section: 4
Variables: \(I\) a set,
\(\mathcal H\) a non-empty subset of \(I\)
Notation: \(\mathcal F^{\mathcal H}\)
Index: (3)
Kind: Ex
Definition2: The set of subset \(A\) of \(I\) containing a finite intersections of element of \(\mathcal H\)

Name: Principal filter generated by \(B\)
Definition: \(\mathcal F^{\{B\}}\)
Page: 18
Chapter: 2
Section: 4
Variables: \(I\) a set,
\(B\subseteq I\) a non-empty subset of \(I\)
Index: (3)
Kind: Ex

Smaller0: \(I\) is finite
Greater0: the set of ultrafilter
Page: 18
Chapter: 2
Section: 4
Variables: \(I\) a set,
Index: (1)
Kind: Ex
Prefix Greater: the set of principal ultrafilter equals

Smaller0: \(I\) is infinite
Greater0: proper
Page: 18
Chapter: 2
Section: 4
Variables: \(I\) a set,
Index: (1)
Kind: Ex
Smaller to greater: iff
Prefix Greater: \(\mathcal F^{co}\) is

Smaller0: \(\mathcal F\) is an ultrafilter
Greater0: \(A\cap B\in\mathcal F\)
Page: 19
Chapter: 2
Section: 5
Variables: \(I\) a set,
\(\mathcal F\) a filter on \(I\)
Index: (3)
Kind: Ex
Greater1: \(A\in\mathcal F\) and \(B\in\mathcal F\)
Connect Greater0: iff
Extra: why ultra ?

Smaller0: \(\mathcal F\) is an ultrafilter
Greater0: \(A\cup B\in\mathcal F\)
Page: 19
Chapter: 2
Section: 5
Variables: \(I\) a set,
\(\mathcal F\) a filter on \(I\)
Index: (3)
Kind: Ex
Greater1: \(A\in\mathcal F\) or \(B\in\mathcal F\)
Connect Greater0: iff

Smaller0: \(\mathcal F\) is an ultrafilter
Greater0: iff \(A\in\mathcal F\)
Page: 19
Chapter: 2
Section: 5
Variables: \(I\) a set,
\(\mathcal F\) a filter on \(I\)
Index: (3)
Kind: Ex
Prefix Greater: \(A^c\in\mathcal F\)

Smaller0: \(\mathcal F\) is an ultrafilter
Greater0: \(\bigcup A_i\in\mathcal F\)
Page: 19
Chapter: 2
Section: 5
Variables: \(I\) a set,
\(\mathcal F\) a filter on \(I\)
\(A_1,\dots,A_n\subseteq I\)
Index: (3)
Kind: Ex
Smaller1: \(A_i\) pairwise disjoint
Greater1: exactly one \(A_i\) belongs to \(\mathcal F\)
Connect Greater0: iff

Smaller0: \(\mathcal F\) is an ultrafilter
Greater0: \(\mathcal F\) is principal
Page: 19
Chapter: 2
Section: 5
Variables: \(I\) a set,
\(\mathcal F\) a filter on \(I\)
\(A_1,\dots,A_n\subseteq I\)
Index: (5)
Kind: Ex
Smaller1: \(\mathcal F\) contains a finite set

Smaller0: \(\mathcal F\) is an ultrafilter
Greater0: \(\mathcal F\) contains all cofinite set
Page: 19
Chapter: 2
Section: 5
Variables: \(I\) a set,
\(\mathcal F\) a filter on \(I\)
Index: (5)
Kind: Ex
Smaller1: \(\mathcal F\) is nonprincipal

Name: A collection with the finite intersection property
Definition: A collection \(\mathcal H\subseteq \mathcal P(I)\) such that, \(B_1\cap\dots\cap B_n\) is non-empty for each \(B_1,\dots,B_n\in\mathcal H\)
Page: 19, 20
Chapter: 2
Section: 5, 6
Variables: \(I\) a set
Context: Filter
Index: (7)
Kind: Ex
Example: \(\{I\}\)
Abbreviation: fip

Smaller0: \(\mathcal H\) has the fip
Greater0: \(\mathcal H\cup\{A\}\) or \(\mathcal H\cup\{A^c\}\) has the fip
Page: 19
Chapter: 2
Section: 5
Variables: \(I\) a set
\(A\subseteq I\)
\(\mathcal H\subseteq \mathcal P(I)\)
Index: (8)
Kind: Ex
Smaller to greater: iff
Extra: fip
Context: Filter

Smaller0: has the finite intersection property
Greater0: can be extended to an ultrafilter on \(I\)
Page: 20
Chapter: 2
Section: 6
Variables: \(I\) a set
\(\mathcal H\subseteq\mathcal P(I)\)
Index: 1
Kind: Theo
Prefix Greater: \(\mathcal H\)
Assuming: Zorn

Smaller0: \(I\) is infinite
Greater0: non principal
Page: 20
Chapter: 2
Section: 6
Variables: \(I\) a set
Index: 1, (3)
Kind: Theo, ex
Smaller to greater: iff
Prefix Greater: \(I\) admits an ultrafilter

Name: \(\langle r_n\rangle\) and \(\langle s_n\rangle\) agrees on a large set
Definition: \(E_{rs}\in\mathcal F\)
Page: 24, 24
Chapter: 3
Section: 2, 4
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
\(\langle r_n\rangle,\langle s_n\rangle\in\mathbb{R^N}\)
Notation: \(\langle r_n\rangle\equiv\langle s_n\rangle\)
Kind: Theo
Name2: \(\langle r_n\rangle\) and \(\langle s_n\rangle\) agrees on a large almost everywhere modulo \(\mathcal F\)
Intuition: \(\langle r_n\rangle\) and \(\langle s_n\rangle\) agrees on at almost all \(n\)
Typ: Congruenceon \(\mathbb{R^N}\)

Definition: the set of \(\langle s_n\rangle\in\mathbb{R^N}\) such that \(r\equiv s\)
Page: 24
Chapter: 3
Section: 2
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
\(\langle r_n\rangle\in\mathbb{R^N}\)
Notation: \([r]\)
Kind: Theo

Definition: elements: set of \([r]\) for \(\langle r_n\rangle\in\mathbb{R^N}\)
Page: 25
Chapter: 3
Section: 2
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
Notation: \(^*\mathbb R\)
Index: 1
Kind: Theo
Definition2: pointwise sum and product
Conjdef: The ordered field
Definition3: \([r]<[s]\) iff \([[r<s]]\in\mathcal F\)
Definition4: 0 \(\langle 0\rangle\)
Assuming: \(\mathcal F\) nonprincipal
Definition5: unity \(\langle 1\rangle\)
Definition6: containing the reals, with \(r\in\mathbb R\) as \(\langle, r,r,\dots\rangle\)

Definition: \(\langle r,r,\dots,\rangle\)
Page: 27
Chapter: 3
Section: 7
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
\(r\in\mathbb R\)
Notation: \(\mathbf r\)
Index: 1
Kind: Theo

Definition: \([\mathbf r]\)
Page: 27
Chapter: 3
Section: 7
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
\(r\in\mathbb R\)
Notation: \(^*r\)
Index: 1
Kind: Theo
Typ: Order preserving field isomorphism from \(\mathbb R\) into \(^*\mathbb R\)

Definition: The set of \([r]\in\mathbb{R^N}\) such that \([[r\in A]]\in\mathcal F\)
Page: 27
Chapter: 3
Section: 7
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
\(A\subseteq\mathbb R\)
Notation: \(^*A\)
Typ: superset of \(A\)

Name: A nonstandard element of \(^*A\)
Definition: Element of \(^*A\setminus A\)
Page: 29
Chapter: 3
Section: 7
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
\(A\subseteq\mathbb R\)
Name2: Ideal member of \(^*A\)
Assuming: \(\mathcal F\) non principal

Smaller0: \(A\) is infinite
Greater0: \(^*A\) has non standard member
Page: 29
Chapter: 3
Section: 9
Variables: \(\mathcal F\) a fixed ultrafilter on \(\mathbb N\)
\(A\subseteq\mathbb R\)
Index: 1
Kind: Theo
Smaller to greater: iff
Assuming: \(\mathcal F\) nonprincipal
Context: non standard analysis

Name: Filters on \(I\)
Page: 18
Section: 4, 5
Chapter: 2
Index: (4), 26
Variables: \(I\) a set,
Kind: Ex
Under: union
Condition: linearly ordered by set inclusion
Under2: intersection ?

Smaller0: \(\mathcal H^I\) is a proper filter
Greater0: \(\mathcal H\) has the finite intersection property
Page: 18
Variables: \(I\) a set
\(\mathcal H\subseteq\mathcal P(I)\)
Smaller to greater: iff