Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Convergence along a filter

John Griffin

Texas A&M University

AMS Gig’em, March 2017

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Theorem (Closure using sequences)Let A be a subset of a metric space X and x ∈ X. There existsa sequence in A converging to x iff x ∈ A.

Theorem (Continuity using sequences)A function f : X → Y from a metric space into a topologicalspace is continuous iff xn → x in X implies f(xn)→ f(x) in Y .

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Theorem (Closure using sequences)Let A be a subset of a metric space X and x ∈ X. There existsa sequence in A converging to x iff x ∈ A.

Theorem (Continuity using sequences)A function f : X → Y from a metric space into a topologicalspace is continuous iff xn → x in X implies f(xn)→ f(x) in Y .

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Theorem (Closure using sequences)Let A be a subset of a metric space X and x ∈ X. There existsa sequence in A converging to x iff x ∈ A.

Theorem (Continuity using sequences)A function f : X → Y from a metric space into a topologicalspace is continuous iff xn → x in X implies f(xn)→ f(x) in Y .

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Theorem (Closure using sequences)Let A be a subset of a metric space X and x ∈ X. There existsa sequence in A converging to x iff x ∈ A.

Theorem (Continuity using sequences)A function f : X → Y from a metric space into a topologicalspace is continuous iff xn → x in X implies f(xn)→ f(x) in Y .

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Sequential convergence cannot determine the topology ofan arbitrary topological space.

ExampleA sequence converges with respect to the cocountabletopology iff the sequence is eventually constant.

Idea: Replace the index set N of a sequence by someother set.

But how do we define what it means for a family (xi)i∈I toconverge to a point x for an arbitrary set I?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Sequential convergence cannot determine the topology ofan arbitrary topological space.

ExampleA sequence converges with respect to the cocountabletopology iff the sequence is eventually constant.

Idea: Replace the index set N of a sequence by someother set.

But how do we define what it means for a family (xi)i∈I toconverge to a point x for an arbitrary set I?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Sequential convergence cannot determine the topology ofan arbitrary topological space.

ExampleA sequence converges with respect to the cocountabletopology iff the sequence is eventually constant.

Idea: Replace the index set N of a sequence by someother set.

But how do we define what it means for a family (xi)i∈I toconverge to a point x for an arbitrary set I?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Sequential convergence cannot determine the topology ofan arbitrary topological space.

ExampleA sequence converges with respect to the cocountabletopology iff the sequence is eventually constant.

Idea: Replace the index set N of a sequence by someother set.

But how do we define what it means for a family (xi)i∈I toconverge to a point x for an arbitrary set I?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Sequential convergence cannot determine the topology ofan arbitrary topological space.

ExampleA sequence converges with respect to the cocountabletopology iff the sequence is eventually constant.

Idea: Replace the index set N of a sequence by someother set.

But how do we define what it means for a family (xi)i∈I toconverge to a point x for an arbitrary set I?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Recall that xn → x in topological space X if for everyneighborhood U of x there exists N ∈ N such that xn ∈ Uwhenever n ≥ N .

Equivalently, for every neighborhood U of x the set{n ∈ N | xn ∈ U} is cofinite.

Note that {cofinite subsets of N} consists of all “large”subsets of N.

Given a set I, what properties should a collection of its”large” subsets satisfy?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Recall that xn → x in topological space X if for everyneighborhood U of x there exists N ∈ N such that xn ∈ Uwhenever n ≥ N .

Equivalently, for every neighborhood U of x the set{n ∈ N | xn ∈ U} is cofinite.

Note that {cofinite subsets of N} consists of all “large”subsets of N.

Given a set I, what properties should a collection of its”large” subsets satisfy?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Recall that xn → x in topological space X if for everyneighborhood U of x there exists N ∈ N such that xn ∈ Uwhenever n ≥ N .

Equivalently, for every neighborhood U of x the set{n ∈ N | xn ∈ U} is cofinite.

Note that {cofinite subsets of N} consists of all “large”subsets of N.

Given a set I, what properties should a collection of its”large” subsets satisfy?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Recall that xn → x in topological space X if for everyneighborhood U of x there exists N ∈ N such that xn ∈ Uwhenever n ≥ N .

Equivalently, for every neighborhood U of x the set{n ∈ N | xn ∈ U} is cofinite.

Note that {cofinite subsets of N} consists of all “large”subsets of N.

Given a set I, what properties should a collection of its”large” subsets satisfy?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Theorems using sequencesInadequacy of sequencesGeneralizing cofiniteness

Recall that xn → x in topological space X if for everyneighborhood U of x there exists N ∈ N such that xn ∈ Uwhenever n ≥ N .

Equivalently, for every neighborhood U of x the set{n ∈ N | xn ∈ U} is cofinite.

Note that {cofinite subsets of N} consists of all “large”subsets of N.

Given a set I, what properties should a collection of its”large” subsets satisfy?

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Definition (Filter)A collection F of subsets of I is a filter on I if

(i)(ii)(iii)

Definition (Convergence)

Let (xi)i∈I be a family in a topological space X. Given a filter Fon I and a point x in X, we say (xi)i∈I converges to x along F

and write xi → x along F if

{i ∈ I | xi ∈ U} ∈ F

for every neighborhood U of x.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Definition (Filter)A collection F of subsets of I is a filter on I if

(i) I ∈ F, ∅ 6∈ F,(ii)(iii)

Definition (Convergence)

Let (xi)i∈I be a family in a topological space X. Given a filter Fon I and a point x in X, we say (xi)i∈I converges to x along F

and write xi → x along F if

{i ∈ I | xi ∈ U} ∈ F

for every neighborhood U of x.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Definition (Filter)A collection F of subsets of I is a filter on I if

(i) I ∈ F, ∅ 6∈ F,(ii) A ∈ F, B ⊇ A =⇒ B ∈ F,(iii)

Definition (Convergence)

Let (xi)i∈I be a family in a topological space X. Given a filter Fon I and a point x in X, we say (xi)i∈I converges to x along F

and write xi → x along F if

{i ∈ I | xi ∈ U} ∈ F

for every neighborhood U of x.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Definition (Filter)A collection F of subsets of I is a filter on I if

(i) I ∈ F, ∅ 6∈ F,(ii) A ∈ F, B ⊇ A =⇒ B ∈ F,(iii) A,B ∈ F =⇒ A ∩B ∈ F.

Definition (Convergence)

Let (xi)i∈I be a family in a topological space X. Given a filter Fon I and a point x in X, we say (xi)i∈I converges to x along F

and write xi → x along F if

{i ∈ I | xi ∈ U} ∈ F

for every neighborhood U of x.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Definition (Filter)A collection F of subsets of I is a filter on I if

(i) I ∈ F, ∅ 6∈ F,(ii) A ∈ F, B ⊇ A =⇒ B ∈ F,(iii) A,B ∈ F =⇒ A ∩B ∈ F.

Definition (Convergence)

Let (xi)i∈I be a family in a topological space X. Given a filter Fon I and a point x in X, we say (xi)i∈I converges to x along F

and write xi → x along F if

{i ∈ I | xi ∈ U} ∈ F

for every neighborhood U of x.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Definition (Filter)A collection F of subsets of I is a filter on I if

(i) I ∈ F, ∅ 6∈ F,(ii) A ∈ F, B ⊇ A =⇒ B ∈ F,(iii) A,B ∈ F =⇒ A ∩B ∈ F.

Definition (Convergence)

Let (xi)i∈I be a family in a topological space X. Given a filter Fon I and a point x in X, we say (xi)i∈I converges to x along F

and write xi → x along F if

{i ∈ I | xi ∈ U} ∈ F

for every neighborhood U of x.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Define a filter on an infinite set I by {cofinite subsets of I}.This is called the Frechet filter on I.

RemarkA sequence (xn) converges to a point x iff the family (xn)n∈Nconverges to x along the Frechet filter on N.

Given a nonempty subset S of I, then {supersets of S} iscalled the principle filter generated by S on I.

Let X be a topological space and x0 ∈ X. We define theneighborhood filter at x0 on X by {neighborhoods of x0}.The family (x)x∈X converges to x0 along this filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Define a filter on an infinite set I by {cofinite subsets of I}.This is called the Frechet filter on I.

RemarkA sequence (xn) converges to a point x iff the family (xn)n∈Nconverges to x along the Frechet filter on N.

Given a nonempty subset S of I, then {supersets of S} iscalled the principle filter generated by S on I.

Let X be a topological space and x0 ∈ X. We define theneighborhood filter at x0 on X by {neighborhoods of x0}.The family (x)x∈X converges to x0 along this filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Define a filter on an infinite set I by {cofinite subsets of I}.This is called the Frechet filter on I.

RemarkA sequence (xn) converges to a point x iff the family (xn)n∈Nconverges to x along the Frechet filter on N.

Given a nonempty subset S of I, then {supersets of S} iscalled the principle filter generated by S on I.

Let X be a topological space and x0 ∈ X. We define theneighborhood filter at x0 on X by {neighborhoods of x0}.The family (x)x∈X converges to x0 along this filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Define a filter on an infinite set I by {cofinite subsets of I}.This is called the Frechet filter on I.

RemarkA sequence (xn) converges to a point x iff the family (xn)n∈Nconverges to x along the Frechet filter on N.

Given a nonempty subset S of I, then {supersets of S} iscalled the principle filter generated by S on I.

Let X be a topological space and x0 ∈ X. We define theneighborhood filter at x0 on X by {neighborhoods of x0}.The family (x)x∈X converges to x0 along this filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Define a filter on an infinite set I by {cofinite subsets of I}.This is called the Frechet filter on I.

RemarkA sequence (xn) converges to a point x iff the family (xn)n∈Nconverges to x along the Frechet filter on N.

Given a nonempty subset S of I, then {supersets of S} iscalled the principle filter generated by S on I.

Let X be a topological space and x0 ∈ X. We define theneighborhood filter at x0 on X by {neighborhoods of x0}.

The family (x)x∈X converges to x0 along this filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Define a filter on an infinite set I by {cofinite subsets of I}.This is called the Frechet filter on I.

RemarkA sequence (xn) converges to a point x iff the family (xn)n∈Nconverges to x along the Frechet filter on N.

Given a nonempty subset S of I, then {supersets of S} iscalled the principle filter generated by S on I.

Let X be a topological space and x0 ∈ X. We define theneighborhood filter at x0 on X by {neighborhoods of x0}.The family (x)x∈X converges to x0 along this filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Theorem (Hausdorff spaces have unique limits)A topological space is Hausdorff iff every family converges to atmost one point along any filter.

Theorem (Closure using convergence along a filter)Let A be a subset of a topological space X and x ∈ X. Thereexists a family (xi)i∈I in A which converges to x along somefilter on I iff x ∈ A.

Theorem (Continuity using convergence along a filter)Let X and Y be topological spaces. A function f : X → Y iscontinuous iff whenever a family (xi)i∈I converges to a point xalong a filter F on I, then f(xi)→ f(x) along F.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Theorem (Hausdorff spaces have unique limits)A topological space is Hausdorff iff every family converges to atmost one point along any filter.

Theorem (Closure using convergence along a filter)Let A be a subset of a topological space X and x ∈ X. Thereexists a family (xi)i∈I in A which converges to x along somefilter on I iff x ∈ A.

Theorem (Continuity using convergence along a filter)Let X and Y be topological spaces. A function f : X → Y iscontinuous iff whenever a family (xi)i∈I converges to a point xalong a filter F on I, then f(xi)→ f(x) along F.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Theorem (Hausdorff spaces have unique limits)A topological space is Hausdorff iff every family converges to atmost one point along any filter.

Theorem (Closure using convergence along a filter)Let A be a subset of a topological space X and x ∈ X. Thereexists a family (xi)i∈I in A which converges to x along somefilter on I iff x ∈ A.

Theorem (Continuity using convergence along a filter)Let X and Y be topological spaces. A function f : X → Y iscontinuous iff whenever a family (xi)i∈I converges to a point xalong a filter F on I, then f(xi)→ f(x) along F.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Filter and convergenceExamplesTheorems

Theorem (Hausdorff spaces have unique limits)A topological space is Hausdorff iff every family converges to atmost one point along any filter.

Theorem (Closure using convergence along a filter)Let A be a subset of a topological space X and x ∈ X. Thereexists a family (xi)i∈I in A which converges to x along somefilter on I iff x ∈ A.

Theorem (Continuity using convergence along a filter)Let X and Y be topological spaces. A function f : X → Y iscontinuous iff whenever a family (xi)i∈I converges to a point xalong a filter F on I, then f(xi)→ f(x) along F.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

Neither direction holds in general.

If F and G are two filters on I such that F ⊆ G, then xi → xalong F implies xi → x along G.

Every filter F is a subset of a maximal filter.

Definition (Ultrafilter)An ultrafilter is a maximal filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

Neither direction holds in general.

If F and G are two filters on I such that F ⊆ G, then xi → xalong F implies xi → x along G.

Every filter F is a subset of a maximal filter.

Definition (Ultrafilter)An ultrafilter is a maximal filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

Neither direction holds in general.

If F and G are two filters on I such that F ⊆ G, then xi → xalong F implies xi → x along G.

Every filter F is a subset of a maximal filter.

Definition (Ultrafilter)An ultrafilter is a maximal filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

Neither direction holds in general.

If F and G are two filters on I such that F ⊆ G, then xi → xalong F implies xi → x along G.

Every filter F is a subset of a maximal filter.

Definition (Ultrafilter)An ultrafilter is a maximal filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

Neither direction holds in general.

If F and G are two filters on I such that F ⊆ G, then xi → xalong F implies xi → x along G.

Every filter F is a subset of a maximal filter.

Definition (Ultrafilter)An ultrafilter is a maximal filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Theorem (Compactness using sequences)A metric space is compact iff every sequence has a convergentsubsequence.

Neither direction holds in general.

If F and G are two filters on I such that F ⊆ G, then xi → xalong F implies xi → x along G.

Every filter F is a subset of a maximal filter.

Definition (Ultrafilter)An ultrafilter is a maximal filter.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Proposition (Ultrafilters are decisive)A filter F on I is an ultrafilter iff A ∈ F or Ac ∈ F for every A ⊆ I.

Theorem (Compactness using convergence along an ultrafilter)

A topological space is compact iff every family (xi)i∈Iconverges along every ultrafilter on I.

Corollary (Tychonoff’s theorem)The product of compact spaces is compact.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Proposition (Ultrafilters are decisive)A filter F on I is an ultrafilter iff A ∈ F or Ac ∈ F for every A ⊆ I.

Theorem (Compactness using convergence along an ultrafilter)

A topological space is compact iff every family (xi)i∈Iconverges along every ultrafilter on I.

Corollary (Tychonoff’s theorem)The product of compact spaces is compact.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Proposition (Ultrafilters are decisive)A filter F on I is an ultrafilter iff A ∈ F or Ac ∈ F for every A ⊆ I.

Theorem (Compactness using convergence along an ultrafilter)

A topological space is compact iff every family (xi)i∈Iconverges along every ultrafilter on I.

Corollary (Tychonoff’s theorem)The product of compact spaces is compact.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Proposition (Ultrafilters are decisive)A filter F on I is an ultrafilter iff A ∈ F or Ac ∈ F for every A ⊆ I.

Theorem (Compactness using convergence along an ultrafilter)

A topological space is compact iff every family (xi)i∈Iconverges along every ultrafilter on I.

Corollary (Tychonoff’s theorem)The product of compact spaces is compact.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

DefinitionTheoremAnother application: Non-standard analysis

Let U be an ultrafilter on N containing the Frechet filter.

Define an order ≤ on the field RN by (xn) ≤ (yn) if{n ∈ N | xn ≤ yn} ∈ U.

This gives an equivalence relation ∼ defined by(xn) ∼ (yn) if (xn) ≤ (yn) and (yn) ≤ (xn).

The quotient ∗R := RN/ ∼ is called the hyperreal numbers.

Embed R into ∗R via x 7→ [(x, x, . . .)].

The positive hyperreal number ε := [(1, 1/2, 1/3, . . .)]satisfies ε < 1/n for every n ∈ N.

See Lectures on the Hyperreals by R. Goldblatt and ElementaryCalculus: An Infinitesimal Approach by H. J. Keisler.

John Griffin Convergence along a filter

Motivation for filtersDefinitions, examples, and theroems

Ultrafilters

Thank you!

John Griffin Convergence along a filter