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$X\in\mathbb{F}$

The intersection of any two elements of $\mathbb{F}$ is an element of $\mathbb{F}$.

$\emptyset\notin\mathbb{F}$ (some authors do not include this axiom in the definition of filter)

If $F\in\mathbb{F}$ and $F\subset G\subset X$ then $G\in\mathbb{F}$.
The first two axioms can be replaced by one:

Any finite intersection of elements of $\mathbb{F}$ is an element of $\mathbb{F}$.
with the usual understanding that the intersection of an empty family of subsets of $X$ is the whole set $X$.
A filter $\mathbb{F}$ is said to be fixed or principal if there is $F\in\mathbb{F}$ such that no proper subset of $F$ belongs to $\mathbb{F}$. In this case, $\mathbb{F}$ consists of all subsets of $X$ containing $F$, and $F$ is called a principal element of $\mathbb{F}$. If $\mathbb{F}$ is not principal, it is said to be nonprincipal or free.
If $x$ is any point (or any subset) of any topological space $X$, the set $\mathcal{N}_{x}$ of neighbourhoods of $x$ in $X$ is a filter, called the neighbourhood filter of $x$. If $\mathbb{F}$ is any filter on the space $X$, $\mathbb{F}$ is said to converge to $x$, and we write $\mathbb{F}\to x$, if $\mathcal{N}_{x}\subset\mathbb{F}$. If every neighbourhood of $x$ meets every set of $\mathbb{F}$, then $x$ is called an accumulation point or cluster point of $\mathbb{F}$.
Remarks: The notion of filter (due to H. Cartan) has a simplifying effect on various proofs in analysis and topology. Tychonoffβs theorem would be one example. Also, the two kinds of limit that one sees in elementary real analysis β the limit of a sequence at infinity, and the limit of a function at a point β are both special cases of the limit of a filter: the FrΓ©chet 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.
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Comments
How can it replace the 2 first axioms
"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}$"
Re: How can it replace the 2 first axioms
> "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 $\mathbb{F}$.
Re: How can it replace the 2 first axioms
OK.
Thank you.
WikiPedia defines differently
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 Γ’β°Β€ 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 nonprincipal.
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
Re: WikiPedia defines differently
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.
Re: WikiPedia defines differently
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 (Ae;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
Re: WikiPedia defines differently
> 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
Re: WikiPedia defines differently
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
Filter over "something"
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"
Re: Filter over "something"
sorry.. disregard that.. I made a little mistake.. I don't know how posts can be deleted.. so please disregard it.
Re: Filter over "something"
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!