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filter

Defines: 
principal filter, nonprincipal filter, non-principal filter, free filter, fixed filter, neighbourhood filter, principal element, convergent filter
Keywords: 
topology, set theory
Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

03E99 no label found54A99 no label found

Comments

"The first two axioms can be replaced by one:
Any finite intersection of elements of $ \mathbb{F}$ is an element of $ \mathbb{F}$."

How can this substitute the first axiom?
"$ X\in\mathbb{F}$"

> "The first two axioms can be replaced by one:
> Any finite intersection of elements of $ \mathbb{F}$ is an
> element of $ \mathbb{F}$."
>
> How can this substitute the first axiom?
> "$ X\in\mathbb{F}$"

If $X$ is viewed as the universe, then $X$ is the intersection
of zero elements of $\mathbb{F}$.

OK.

Thank you.

WikiPedia defines principal filter differently:

http://en.wikipedia.org/wiki/Filter_(mathematics)
The smallest filter that contains a given element p is a principal filter and p is a principal element in this situation. The principal filter for p is just given by the set {x in P | p Ò‰€ x} and is denoted by prefixing p with an upward arrow.

But PlanetMath defines:
A filter $ \mathbb{F}$ is said to be fixed or principal if the intersection of all elements of $\mathbb{F}$ is nonempty; otherwise, $\mathbb{F}$ is said to be free or non-principal.

Every filter principal in the sense of WikiPedia is principal in the sense of PlanetMath, but not vice verse.

We need to resolve this terminological issue.
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts

Your issue is probably best filed in form of a correction.

Do you have an example of a filter which is principal
according to one definition but not according to the
other? (I'm no expert on filters, so it would be interesting
to see, regardless of how the issue is resolved.) If not,
then there's a fair chance IMHO that the two definitions
actually are equivalent.

Let there are a line L and a point A on this line ($A\in L$).
Consider the filter which consists of all subsets of a line having A as their internal point (that is all sets which contain some interval (A-e;A+e).

This is:
principal filter accordingly PlanetMath
not a principal filter accordingly WikiPedia.
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts

> Your issue is probably best filed in form of a correction.

I would file a correction if would be sure what is correct. I suspect that WikiPedia is correct and PlanetMath mistakes on this issue, but I'm not 100% sure and so can't file a correction.
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts

WikiPedia discusses `principal', `fixed', or `trivial' (all three terms considered equivalent) also at the following URL:
http://en.wikipedia.org/wiki/Ultrafilter

This confirms that right is the other definition in WikiPedia
http://en.wikipedia.org/wiki/Filter_%28mathematics%29

I will file a correction for PlanetMath's article.
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts

When we refer to a filter, we may regard it as a filter over a subfamily of the powerset of X, and not necessarily over the powerset of X. See "multiplicative filters"

sorry.. disregard that.. I made a little mistake.. I don't know how posts can be deleted.. so please disregard it.

I will try to forget about it, but I'm not sure how much more
I'll have to forget -- this could result in a total mindwipe!
Gulp! Here goes!

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