aarjaneiro/planetmath huggingface dataset:
axiom of determinacy

axiom of determinacy

When doing

descriptive set theory

, it is conventional to use either

ω

ω

or

2

ω

as your space of “reals” (these spaces are

homeomorphic

to the irrationals and the

Cantor set

, respectively). Throughout this article, I will use the term “reals” to refer to

ω

ω

.

Let

X

⊆

ω

ω

be given and consider the following game on

X

played between two players, I and II: I starts by saying a

natural number

; II hears this number and replies with another (or possibly the same one); I hears this and replies with another; etc. The

sequence

of numbers said (in the order they were said) is a point in

ω

ω

. I wins if this point is in

X

, otherwise II wins.

A map

σ

:

ω

<

ω

→

ω

is said to be a winning strategy for I if it has the following property: if, after the play has gone

n

0

⁢

n

1

⁢

…

⁢

n

M

, I plays

σ

⁢

(

n

0

⁢

…

⁢

n

M

)

for each move, then I wins. A winning strategy for II is defined analogously.

The

axiom of determinacy

(AD) states that every such game is determined, that is either I or II has a winning strategy.

Using choice, a non-determined game can be constructed directly: for

α

<

𝔠

, enumerate the

uncountable

closed subsets

of the reals

F

α

. Now

construct

two sequences

⟨

x

α

:

α

<

𝔠

⟩

and

⟨

y

α

:

α

<

𝔠

⟩

by choosing

x

α

,

y

α

as distinct points from

F

α

which are not in

{

x

γ

,

y

γ

:

γ

<

α

}

(this is possible as each uncountable

closed set

has

cardinality

𝔠

). Then the game on the set of all

x

α

s is non-determined.

From ZF+AD, one may prove many nice facts about the reals, such as: any subset is

Lebesgue measurable

, any subset has a

perfect

subset and the

continuum hypothesis

. ZF+AD also proves the

axiom of countable choice

.

AD itself is not taken seriously by many set theorists as a genuine alternative to choice. However, there is a weakening of AD (the axiom of quasi-projective determinacy, or QPD, which states that all games in

𝖫

⁢

[

ℝ

]

are determined) which is

consistent

with ZFC (in fact, it’s equiconsistent to a large cardinal axiom) which is a serious axiom candidate.

Title

axiom of determinacy

Canonical name

AxiomOfDeterminacy

Date of creation

2013-03-22 14:50:46

Last modified on

2013-03-22 14:50:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Axiom

Classification

msc 03E60

Classification

msc 03E15

Synonym

AD

Generated on Thu Feb 8 20:20:22 2018 by

LaTeXML
combinatorial principle

combinatorial principle

A

combinatorial principle

is any statement

Φ

of

set theory

proved to be

independent

of Zermelo-Fraenkel (ZF) set theory, usually one with interesting

consequences

.

If

Φ

is a combinatorial principle, then whenever we have

implications

of the form

P

⟹

Φ

⟹

Q

,

we automatically know that

P

is unprovable in ZF and

Q

is relatively

consistent

with ZF.

Some examples of combinatorial principles are the

axiom of choice

(

http://planetmath.org/AxiomOfChoice

), the

continuum hypothesis

,

◇

,

♣

, and Martin’s axiom.

References

1

Just, W.,

http://www.math.ohiou.edu/ just/resint.html#principles

http://www.math.ohiou.edu/~just/resint.html#principles.

Title

combinatorial principle

Canonical

name

CombinatorialPrinciple

Date of creation

2013-03-22 14:17:41

Last modified on

2013-03-22 14:17:41

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Definition

Classification

msc 03E65

Related topic

Diamond

Related topic

Clubsuit

Generated on Thu Feb 8 20:20:24 2018 by

LaTeXML
◇

◇

Definition 1

.

Let

S

⊆

κ

be a

stationary set

. Then the

combinatorial principle

◇

S

holds if and only if there is a

sequence

⟨

A

α

⟩

α

∈

S

such that each

A

α

⊆

α

and for any

A

⊆

κ

,

{

α

∈

S

∣

A

∩

α

=

A

α

}

is stationary.

To get some sense of what this means, observe that for any

λ

<

κ

,

{

λ

}

⊆

κ

, so the set of

A

α

=

{

λ

}

is stationary (in

κ

). More strongly, suppose

κ

>

λ

. Then any subset of

T

⊂

λ

is bounded in

κ

so

A

α

=

T

on a stationary set. Since

|

S

|

=

κ

, it follows that

2

λ

≤

κ

. Hence

◇

ℵ

1

, the most common form (often written as just

◇

), implies CH.

C. Akemann and N. Weaver used

◇

to construct a

C

*

-

algebra

serving as a

counterexample

to Naimark’s problem.

References

1

Akemann, C., and N. Weaver,

Consistency

of a counterexample to Naimark’s problem

. Preprint available on the arXiv at

http://arxiv.org/abs/math.OA/0312135

http://arxiv.org/abs/math.OA/0312135.

Title

◇

Canonical

name

Diamond

Date of creation

2013-03-22 12:53:49

Last modified on

2013-03-22 12:53:49

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 03E65

Synonym

diamond

Related topic

Clubsuit

Related topic

DiamondIsEquivalentToClubsuitAndContinuumHypothesis

Related topic

ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis

Related topic

CombinatorialPrinciple

Defines

◇

S

Generated on Thu Feb 8 20:20:26 2018 by

LaTeXML
♣

♣

♣

S

is a

combinatoric

principle weaker than

◇

S

. It states that, for

S

stationary

in

κ

, there is a

sequence

⟨

A

α

⟩

α

∈

S

such that

A

α

⊆

α

and

sup

⁡

(

A

α

)

=

α

and with the

property

that for each

unbounded

subset

T

⊆

κ

there is some

A

α

⊆

X

.

Any sequence satisfying

◇

S

can be adjusted so that

sup

⁡

(

A

α

)

=

α

, so this is indeed a weakened form of

◇

S

.

Any such sequence actually contains a stationary set of

α

such that

A

α

⊆

T

for each

T

: given any club

C

and any unbounded

T

, construct a

κ

sequence,

C

*

and

T

*

, from the elements of each, such that the

α

-th member of

C

*

is greater than the

α

-th member of

T

*

, which is in turn greater than any earlier member of

C

*

. Since both sets are unbounded, this construction is possible, and

T

*

is a subset of

T

still unbounded in

κ

. So there is some

α

such that

A

α

⊆

T

*

, and since

sup

⁡

(

A

α

)

=

α

,

α

is also the limit of a

subsequence

of

C

*

and therefore an element of

C

.

Title

♣

Canonical

name

clubsuit

Date of creation

2013-03-22 12:53:52

Last modified on

2013-03-22 12:53:52

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03E65

Synonym

clubsuit

Related topic

Diamond

Related topic

DiamondIsEquivalentToClubsuitAndContinuumHypothesis

Related topic

ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis

Related topic

CombinatorialPrinciple

Generated on Thu Feb 8 20:20:28 2018 by

LaTeXML
paradox of the binary tree

paradox of the binary tree

The

complete infinite binary tree

is a tree that consists of nodes (namely the numerals 0 and 1) such that every node has two

children

which are not children of any other node. The tree serves as binary

representation

of all

real numbers

of the

interval

[0,1] in form of paths, i.e.,

sequences

of nodes.

Every finite

binary tree

with more than one level contains less paths than nodes. Up to level n there are 2^n paths and 2^(n+1) - 1 nodes.

Every finite binary tree can be represented as an ordered set of nodes, enumerated by

natural numbers

. The union of all finite binary trees is then identical with the

infinite

binary tree. The

paradox

is that, while the set of nodes
remains

countable

as is the set of paths of all finite trees, the set of paths in the infinite tree is

uncountable

by Cantor’s

theorem

. (On the other hand, the paths are

separated

by the nodes. As no path can separate itself from another path without a node, the number of separated paths is the number of nodes.)

Literature: W. Mückenheim: Die Mathematik des Unendlichen, Shaker-Verlag, Aachen 2006.

Title

paradox of the binary tree

Canonical name

ParadoxOfTheBinaryTree

Date of creation

2013-03-22 17:02:37

Last modified on

2013-03-22 17:02:37

Owner

WM (16977)

Last modified by

WM (16977)

Numerical id

21

Author

WM (16977)

Entry type

Definition

Classification

msc 03E75

Classification

msc 03E15

Synonym

binary tree paradox

Defines

complete binary tree

Defines

complete infinite binary tree

Generated on Thu Feb 8 20:20:36 2018 by

LaTeXML
alternative characterization of filter

alternative characterization of filter

A

collection

𝐅

of subsets of a set

X

is a filter if and only if the following condition is satisfied: For any two subsets

A

and

B

of

X

, it is the case that

A

∩

B

∈

𝐅

if and only if

A

∈

𝐅

and

B

∈

𝐅

.

Title

alternative characterization of filter

Canonical name

AlternativeCharacterizationOfFilter

Date of creation

2013-03-22 14:43:02

Last modified on

2013-03-22 14:43:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03E99

Classification

msc 54A99

Generated on Thu Feb 8 20:20:37 2018 by

LaTeXML
basic properties of a limit along a filter

basic properties of a limit along a filter

Theorem 1

.

Let

F

be a

free filter

(non-principal filter) and

(

x

n

)

be a real sequence.

(i)

If

lim

n

→

∞

⁡

x

n

=

L

then

ℱ

⁢

-

⁢

lim

⁡

x

n

=

L

.

(ii)

If

ℱ

⁢

-

⁢

lim

⁡

x

n

exists, then

lim inf

⁡

x

n

≤

ℱ

⁢

-

⁢

lim

⁡

x

n

≤

lim sup

⁡

x

n

.

(iii)

The

ℱ

-limits are unique.

(iv)

ℱ

⁢

-

⁢

lim

(

a

.

x

n

+

b

.

y

n

)

=

a

.

ℱ

⁢

-

⁢

lim

x

n

+

b

.

ℱ

⁢

-

⁢

lim

y

n

(

provided the

ℱ

-limits of

(

x

n

)

and

(

y

n

)

exist

)

.

(v)

ℱ

⁢

-

⁢

lim

(

x

n

.

y

n

)

=

ℱ

⁢

-

⁢

lim

x

n

.

ℱ

⁢

-

⁢

lim

y

n

(

provided the

ℱ

-limits of

(

x

n

)

and

(

y

n

)

exist

)

.

(vi)

For every

cluster point

c

of the sequence

x

n

there exists
a free filter

ℱ

such that

ℱ

⁢

-

⁢

lim

⁡

x

n

=

c

. On the other hand, if

ℱ

⁢

-

⁢

lim

⁡

x

n

exists, it is a cluster point of the sequence

(

x

n

)

.

Title

basic properties of a limit along a filter

Canonical name

BasicPropertiesOfALimitAlongAFilter

Date of creation

2013-03-22 15:32:23

Last modified on

2013-03-22 15:32:23

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

9

Author

kompik (10588)

Entry type

Theorem

Classification

msc 03E99

Classification

msc 40A05

Generated on Thu Feb 8 20:20:40 2018 by

LaTeXML
collection

collection

The term

collection

usually denotes a class of sets (as opposed to a set of

elements

).

The term “

collection

of sets” is a synonym for “class of sets”.

Title

collection

Canonical name

Collection

Date of creation

2013-03-22 16:35:43

Last modified on

2013-03-22 16:35:43

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

6

Author

porton (9363)

Entry type

Definition

Classification

msc 03E99

Synonym

SetOfSets

Related topic

Set

Related topic

Selector

Generated on Thu Feb 8 20:20:44 2018 by

LaTeXML
complement

complement

1

Definition

Let

A

be a subset of

X

. The

complement

of

A

in

X

(denoted

A

∁

when the larger set

X

is clear from

context

) is the

set difference

X

∖

A

.

The

Venn diagram

below illustrates the complement of

A

in red.

2

Properties

•

(

A

∁

)

∁

=

A

•

∅

∁

=

X

•

X

∁

=

∅

•

If

A

and

B

are subsets of

X

, then

A

∖

B

=

A

∩

B

∁

, where the complement is taken in

X

.

3

de Morgan’s laws

Let

X

be a set with subsets

A

i

⊂

X

for

i

∈

I

, where

I

is an arbitrary index-set. In other words,

I

can be finite,

countable

, or

uncountable

. Then

(

⋃

i

∈

I

A

i

)

∁

=

⋂

i

∈

I

A

i

∁

,

(

⋂

i

∈

I

A

i

)

∁

=

⋃

i

∈

I

A

i

∁

.

Title

complement

Canonical

name

Complement

Date of creation

2013-03-22 12:18:51

Last modified on

2013-03-22 12:18:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E99

Related topic

DeMorgansLaws

Generated on Thu Feb 8 20:20:46 2018 by

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cumulative hierarchy

cumulative hierarchy

The

cumulative hierarchy

of sets
is defined by

transfinite recursion

as follows:
we define

V

0

=

∅

and for each

ordinal

α

we define

V

α

+

1

=

𝒫

⁢

(

V

α

)

and for each

limit ordinal

δ

we define

V

δ

=

⋃

α

∈

δ

V

α

.

Every set is a subset of

V

α

for some ordinal

α

,
and the least such

α

is called the

rank

of the set.
It can be shown that the rank of an ordinal is itself,
and in general the rank of a set

X

is the least ordinal greater than the rank of every element of

X

.
For each ordinal

α

,
the set

V

α

is the set of all sets of rank less than

α

,
and

V

α

+

1

∖

V

α

is the set of all sets of rank

α

.

Note that the previous paragraph makes use of the

Axiom of Foundation

:
if this axiom fails,
then there are sets that are not subsets of any

V

α

and therefore have no rank.
The previous paragraph also assumes that we are using a

set theory

such as ZF,
in which elements of sets are themselves sets.

Each

V

α

is a

transitive set

.
Note that

V

0

=

0

,

V

1

=

1

and

V

2

=

2

,
but for

α

>

2

the set

V

α

is never an ordinal,
because it has the

element

{

1

}

, which is not an ordinal.

Title

cumulative hierarchy

Canonical name

CumulativeHierarchy

Date of creation

2013-03-22 16:18:43

Last modified on

2013-03-22 16:18:43

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 03E99

Synonym

iterative hierarchy

Synonym

Zermelo hierarchy

Related topic

CriterionForASetToBeTransitive

Related topic

ExampleOfUniverse

Defines

rank

Defines

rank of a set

Generated on Thu Feb 8 20:20:54 2018 by

LaTeXML
Dedekind-infinite

Dedekind-infinite

A set

A

is said to be

Dedekind-infinite

if there is an

injective function

f

:

ω

→

A

,
where

ω

denotes the set of natural numbers.
A set that is not Dedekind-infinite is said to be

Dedekind-finite

.

A Dedekind-infinite set is clearly

infinite

,
and in ZFC it can be shown that
a set is Dedekind-infinite if and only if it is infinite.

It is

consistent

with ZF that
there is an

infinite set

that is not Dedekind-infinite.
However, the

existence

of such a set requires the failure
not just of the full

Axiom of Choice

, but even of the

Axiom of Countable Choice

.

Title

Dedekind-infinite

Canonical name

Dedekindinfinite

Date of creation

2013-03-22 12:05:25

Last modified on

2013-03-22 12:05:25

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 03E99

Synonym

Dedekind infinite

Related topic

Cardinality

Defines

Dedekind-finite

Defines

Dedekind finite

Generated on Thu Feb 8 20:20:55 2018 by

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delta system

delta system

If

S

is a set of

finite sets

then it is a

Δ

-system

if there is some (possibly empty)

X

such that for any

a

,

b

∈

S

, if

a

≠

b

then

a

∩

b

=

X

.

Title

delta system

Canonical

name

DeltaSystem

Date of creation

2013-03-22 12:54:57

Last modified on

2013-03-22 12:54:57

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E99

Related topic

DeltaSystemLemma

Generated on Thu Feb 8 20:20:57 2018 by

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delta system lemma

delta system lemma

If

S

is a set of

finite sets

such that

|

S

|

=

ℵ

1

then there is a

S

′

⊆

S

such that

|

S

′

|

=

ℵ

1

and

S

is a

http://planetmath.org/node/DeltaSystem

Δ

-system.

Title

delta system lemma

Canonical name

DeltaSystemLemma

Date of creation

2013-03-22 12:55:00

Last modified on

2013-03-22 12:55:00

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Theorem

Classification

msc 03E99

Related topic

DeltaSystem

Generated on Thu Feb 8 20:20:59 2018 by

LaTeXML
diagonal intersection

diagonal intersection

If

⟨

S

i

⟩

i

<

α

is a

sequence

then the

diagonal intersection

,

Δ

i

<

α

⁢

S

i

is defined to be

{

β

⁢

<

α

∣

⁢

β

∈

⋂

γ

<

β

S

γ

}

.

That is,

β

is in

Δ

i

<

α

⁢

S

i

if it is

contained

in the first

β

members of the sequence.

Title

diagonal intersection

Canonical name

DiagonalIntersection

Date of creation

2013-03-22 12:53:17

Last modified on

2013-03-22 12:53:17

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03E99

Defines

diagonal intersection

Generated on Thu Feb 8 20:21:01 2018 by

LaTeXML
disjoint union

disjoint union

Disjoint union of two sets

Let

A

and

B

be sets. Then their

disjoint union

is the union

A

⁢

∐

B

:=

A

∪

(

B

×

{

⋆

}

)

,

where

⋆

is an object chosen so that

A

and

B

×

{

⋆

}

are

disjoint

.
Normally

B

×

{

⋆

}

is identified with

B

in the obvious way.
The

element

⋆

is almost never mentioned;
it serves only as a “tag” to make the two sets disjoint.
If

A

and

B

are already disjoint,
then

A

⁢

∐

B

is

isomorphic

to

A

∪

B

;
this is the most common situation in practice.

Disjoint union of many sets

If we have a

collection

of sets

{

A

i

}

indexed by

some set

I

,
then the disjoint union

∐

i

∈

I

A

i

:=

⋃

i

∈

I

A

i

×

{

i

}

.

Observe that we have a

natural isomorphism

A

i

→

A

i

×

{

i

}

,
and that the images of any pair of these

isomorphisms

have empty

intersection

.
This is also often called being

pairwise disjoint

and is a much

stronger

condition than that
the intersection of all the images is empty.
As before, if the

A

i

are already pairwise disjoint, then

∐

i

∈

I

A

i

≅

⋃

i

∈

I

A

i

.

Explanation

If one is working in some

category

,
the term “disjoint union” often means “

coproduct

”.

For example, as sets,

ℝ

⁢

∐

ℝ

is two copies of the real line.
As

topological spaces

,

ℝ

⁢

∐

ℝ

is again two copies of the real line
with a

topology

whose

open sets

are pairs of real open sets,
one for each copy of

ℝ

.
This is the coproduct in the category of topological spaces.

Of course, there are many categories where this usage is unnatural.
For example, in the category of

pointed sets

,
the coproduct is the disjoint union
with the distinguished points identified.
In the category of abelian groups, the coproduct is the

direct sum

.

Another closely related usage should be mentioned.
Occasionally an author will write
“…and

A

∪

B

is a disjoint union…”.
What this means is that

A

∪

B

is isomorphic to

A

⁢

∐

B

,
which is to say that

A

and

B

are already disjoint.

Title

disjoint union

Canonical name

DisjointUnion

Date of creation

2013-03-22 14:13:24

Last modified on

2013-03-22 14:13:24

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 03E99

Related topic

CardinalityOfDisjointUnionOfFiniteSets

Generated on Thu Feb 8 20:21:03 2018 by

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filter

filter

Let

X

be a set. A filter on

X

is a set

𝔽

of subsets of

X

such that

•

X

∈

𝔽

•

The

intersection

of any two elements of

𝔽

is an element of

𝔽

.

•

∅

∉

𝔽

(some authors do not include this axiom in the definition of filter)

•

If

F

∈

𝔽

and

F

⊂

G

⊂

X

then

G

∈

𝔽

.

The first two axioms can be replaced by one:

•

Any finite intersection of elements of

𝔽

is an element of

𝔽

.

with the usual understanding that the intersection of an empty family
of subsets of

X

is the whole set

X

.

A filter

𝔽

is said to be

fixed

or

principal

if there is

F

∈

𝔽

such that no

proper subset

of

F

belongs to

𝔽

. In this case,

𝔽

consists of all subsets of

X

containing

F

, and

F

is called a

principal element

of

𝔽

. If

𝔽

is not principal, it is said to be

non-principal

or

free

.

If

x

is any point (or any subset) of any

topological space

X

,
the set

𝒩

x

of

neighbourhoods

of

x

in

X

is a filter,
called the

neighbourhood filter

of

x

.
If

𝔽

is any filter on the space

X

,

𝔽

is said to

converge

to

x

, and we write

𝔽

→

x

,
if

𝒩

x

⊂

𝔽

.
If every neighbourhood of

x

meets every set of

𝔽

, then

x

is called an

accumulation point

or

cluster point

of

𝔽

.

Remarks:

The notion of filter (due to H. Cartan) has a simplifying effect on
various proofs in

analysis

and

topology

.

Tychonoff

’s

theorem

would be one example.
Also, the two kinds of limit that one sees in

elementary

real
analysis

– the limit of a

sequence

at

infinity

, and the limit
of a

function

at a point – are both special cases of the limit
of a filter: the Fréchet filter and the neighbourhood filter
respectively.
The notion of a

Cauchy sequence

can be extended with no difficulty
to any

uniform space

(but not just a topological space),
getting what is called a Cauchy filter; any convergent filter on a
uniform space is a Cauchy filter, and if the

converse

holds then
we say that the uniform space is

complete

.

Title

filter

Canonical name

Filter

Date of creation

2013-03-22 12:09:06

Last modified on

2013-03-22 12:09:06

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

19

Author

Koro (127)

Entry type

Definition

Classification

msc 03E99

Classification

msc 54A99

Related topic

Ultrafilter

Related topic

KappaComplete

Related topic

KappaComplete2

Related topic

Net

Related topic

LimitAlongAFilter

Related topic

UpperSet

Related topic

OrderIdeal

Defines

principal filter

Defines

nonprincipal filter

Defines

non-principal filter

Defines

free filter

Defines

fixed filter

Defines

neighbourhood filter

Defines

principal element

Defines

convergent filter

Generated on Thu Feb 8 20:21:05 2018 by

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filter basis

filter basis

A

filter subbasis

for a set

S

is a

collection

of subsets of

S

which has the

finite intersection property

.

A

filter basis

B

for a set

S

is a non-empty collection of subsets of

S

which does not contain the

empty set

such that, for every

u

∈

B

and every

v

∈

B

, there exists a

w

∈

B

such that

w

⊂

u

∩

v

.

Given a filter basis

B

for a set

S

, the set of all

supersets

of

elements of

B

forms a filter on the set

S

. This filter is known as the

filter generated by

the basis.

Given a filter subbasis

B

for a set

S

, the set of all supersets of finite

intersections

of elements of

B

is a filter. This filter is known as the filter generated by the

subbasis

.

Two filter bases are said to be

equivalent

if they generate the same filter. Likewise, two filter subbases are said to be equivalent if they generate the same filter.

Note:

Not every author requires that filters do not contain the empty set.
Because every filter is a filter basis then accordingly some authors allow that a filter
base can contain the empty set.

Title

filter basis

Canonical name

FilterBasis

Date of creation

2013-03-22 14:41:34

Last modified on

2013-03-22 14:41:34

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03E99

Classification

msc 54A99

Synonym

filter base

Defines

filter subbasis

Defines

equivalent

Generated on Thu Feb 8 20:21:07 2018 by

LaTeXML