# Questions tagged [filters]

Filters (and ultrafilters) are used in various areas of mathematics, e.g. general topology, set theory, boolean algebras, model theory. In topology they can be used to study convergence in a more general way than just convergence of sequences. For questions about filters in the sense of signal processing, do not use this tag and instead use (signal-processing) or another appropriate tag.

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### The set of ultrafilters on an infinite set is uncountable

After recently learning about filters and ultrafilters, we looked into further problems and properties. I am having trouble with this one: If $X$ is an infinite set, then the set of all ultrafilters ...
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### Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
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### Where has this common generalization of nets and filters been written down?

It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters. They are of course essentially equivalent, ...
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### Applications of ultrafilters

I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters? I'm ...
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### Characterization of weak convergence in $\ell_\infty$

Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets? I was only able to come up with a characterization of ...
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### Ultra Filter and Axiom of Choice

Some person said me: "The fact that Ultra Filters exist is equivalent to the Axiom of choice". Is this correct? I nees some good references about the subject, please help me. Thanks
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### Subnets and finer filters

Suppose $G$ is a finer filter than $F$ in a topological space $X$. Is the net base in $G$ a subnet of the net base in $F$? I am using the definitions of General Topology of Willard: Definition 12....
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### References on filter quantifiers

This post is primarily a reference request. In combinatorics and other areas, we use filter quantifiers to simplify the statements of various definitions, theorems and proofs. The general idea is ...
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### every non-principal ultrafilter contains a cofinite filter.

I have several questions regarding filters and ultrafilters: (a) Does every non-principal ultrafilter contain a cofinite filter? Why? (b) What is the difference between a Fréchet filter and a ...
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### Which "limit of ultrafilter" functions induce a compact Hausdorff topological structure?

Somewhere I saw a brief comment, without proof, that the underlying set functor from compact Hausdorff topological spaces to sets is monadic. So, I found myself wondering if the former category could ...
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### Ultrafilters - when did it start?

I am writing a paper on some of the applications of ultrafilters, especially the ones on $\mathbb{N}$. I thought that it would be interesting to include some information about how the concept got ...
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### Intuition behind filter on a set

In "Counter-examples in topology" of Steen and Seebach, they define a filter on a set $X$ is a collection F of subsets of $X$ with the following properties: Every subset of $X$ which contains a set ...
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### Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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### Ultrafilter Lemma and Alexander subbase theorem

I'm trying to prove the equivalence in ZF between the Ultrafilter Lemma (UF) and the Alexander subbase theorem (AS). Although I've found a way to prove that (AS) $\Rightarrow$ (UF), with the help of ...
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### Addition on ultrafilters is non-commutative

I'm reading a book on ultrafilters and it tells me that showing addition on ultrafilters over the naturals (addition on the set of ultrafilters '$\beta \mathbb{N}$', defined in the standard way) is ...
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### Cardinality of ultraproducts

This might be a trivial question, but I couldn't see how to do this now. Given an infinite model $M$ and infinite cardinal $\kappa > |M|$ and a non-principal ultrafilter $F$ over $\kappa$, is it ...
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### Example of a free ultrafilter on natural numbers

We know that free ultrafilter exist on natural numbers. I would like to see an example of a free ultrafilter on natural numbers.
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### Existence of non-trivial ultrafilter closed under countable intersection

Under what conditions on $\Omega$ does there exist $\mathcal{F} \subset \mathcal{P}(\Omega)$ such that $\mathcal{F}$ is a non-trivial ultrafilter and, for every sequence $(F_{i})_{i \in N}$ of ...
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### Why do we need ultrafilter for construction of hyperreal numbers? [duplicate]

While trying to get some hold on the hyperreal numbers I found that their construction using the already existing real number system $\mathbb{R}$ requires the use of objects called free ultrafilters ...
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### Łoś's Theorem holds for positive sentences at reduced products in general?

Let $\mathcal{L}$ be a language for first-order logic whose logical primitives are $\neg$, $\vee$, $\wedge$, $\forall$, and $\exists$, with the usual formation rules. A sentence $\sigma$ is ...
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### "All ultrafilters are principal" consistent with ZF?

In the article on ultrafilters, Wikipedia claims that In ZF without the axiom of choice, it is possible that every ultrafilter is principal.{see p.316, [Halbeisen, L.J.] "Combinatorial Set Theory", ...
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### Principal ultrafilter and free filter

I have 2 questions about filter and (ultra-)filters: Which relations are there between free filter, principal filter, ultrafilter, Frechet filter, and co-finite filter? If a filter is free, does it ...
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### Can unlimited hypernaturals be represented by increasing sequences?

Consider the usual ultra-power construction of the hyperreals $^*\mathbb R$ with ultrafilter $\mathcal F$. Let $K = [k_n] \in ^*\mathbb N$ be an unlimited hypernatural. My question is does there ...
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### Is an ultrafilter free if and only if it contains the cofinite filter?

A filter $\mathcal F$ is called free if $\bigcap \mathcal F=\emptyset$. Filter, which is not free is called principal. any principal ultrafilter has the form $$\mathcal F_a=\{A\subseteq X; a\in A\}$$ ...
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### H-closed = compact?

This is an exercise from Herrlich's Axiom of choice, and I'm a bit struggling with it. I want to show that a topological space $X$ is compact Hausdorff if and only if it is H-closed and regular. NB :...
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### Club filter of $\kappa$ is $\kappa$-complete

I'm trying to show that club filter of $\kappa$ is $\kappa$-complete for uncountable regular cardinal $\kappa$: Let $\kappa$ be uncountable regular cardinal, let $C(\kappa)$ be the club filter ...
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### Filters vs nets in topology

Nets are a natural generalization of sequences in arbitrary topological spaces. Using the language of nets we can extend intuitive, classical sequential notions (compactness, convergence, etc.) to ...
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### How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
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### The Hausdorff property versus closedness of the diagonal in the context of convergence spaces

Given a topological space $X$, the following are equivalent: Given points $x$ and $y$, there exist neighborhoods $A$ and $B$ of $x$ and $y$ respectively satisfying $A \cap B = \emptyset$. Every ...
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I'm stuck with the next exercise from the book Rings of Continuous Functions by Gillman. If $X$ is infinite, there exist $2^{2^{|X|}}$ ultrafilters on $X$ all of whose members are of cardinal $X$. ...