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 ...
josh's user avatar
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12 votes
1 answer
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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 ...
liman's user avatar
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24 votes
4 answers
2k views

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, ...
Eric Wofsey's user avatar
18 votes
7 answers
3k views

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 ...
Jacob Fox's user avatar
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16 votes
2 answers
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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 ...
Martin Sleziak's user avatar
10 votes
3 answers
3k views

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
Tomás's user avatar
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5 votes
1 answer
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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....
Pedro Perez's user avatar
20 votes
1 answer
753 views

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 ...
Carl Mummert's user avatar
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15 votes
2 answers
1k views

Every multiplicative linear functional on $\ell^{\infty}$ is the limit along an ultrafilter.

It is well-known that for any ultrafilter $\mathscr{u}$ in $\mathbb{N}$, the map\begin{equation}a\mapsto \lim_{\mathscr{u}}a\end{equation} is a multiplicative linear functional, where $\lim_{\mathscr{...
Hui Yu's user avatar
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12 votes
3 answers
1k views

Uniqueness of hyperreals contructed via ultrapowers

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence ...
LCL's user avatar
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10 votes
2 answers
984 views

The non-existence of non-principal ultrafilters in ZF

In Hrbacek and Jech (1999, p.205), they point out that "it is known that the theorem [the extension of any filter to an ultrafilter] cannot be proved in Zermelo-Fraenkel set theory alone." And in Jech ...
Logica's user avatar
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3 votes
2 answers
833 views

Ultrafilter Lemma implies Compactness/Completeness of FOL

Apologies if this has been asked somewhere before, but I didn't see what I was looking for after several pages of Google results. I was reading Jech's The Axiom of Choice and was introduced to the ...
Malice Vidrine's user avatar
8 votes
2 answers
1k views

How can an ultrafilter be considered as a finitely additive measure?

From Wikipedia an ultrafilter $U$ on a set $X$ is a collection of subsets of $X$ that is a filter, that cannot be enlarged (as a filter). An ultrafilter may be considered as a finitely additive ...
Tim's user avatar
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5 votes
2 answers
861 views

Sum and product of ultrafilters

can anyone tell me, please, two ultrafilters such that $\mathcal{U}\otimes\mathcal{V}\neq\mathcal{V}\otimes\mathcal{U}$ and others two such that $\mathcal{U}\oplus\mathcal{V}\neq\mathcal{V}\oplus\...
Jacob Fox's user avatar
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5 votes
1 answer
2k views

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 ...
Tina's user avatar
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3 votes
1 answer
283 views

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 ...
Daniel Schepler's user avatar
19 votes
3 answers
1k views

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 ...
Jakub Konieczny's user avatar
18 votes
3 answers
3k views

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 ...
SiXUlm's user avatar
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11 votes
0 answers
1k views

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 ...
Martin Sleziak's user avatar
10 votes
2 answers
625 views

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 ...
Renan Mezabarba's user avatar
10 votes
1 answer
1k views

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 ...
Lewis Haines's user avatar
9 votes
2 answers
382 views

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 ...
nana's user avatar
  • 337
9 votes
2 answers
3k views

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.
asv's user avatar
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8 votes
1 answer
722 views

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 ...
madprob's user avatar
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8 votes
1 answer
1k views

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 ...
Paramanand Singh's user avatar
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8 votes
1 answer
581 views

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 ...
MikeC's user avatar
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7 votes
1 answer
327 views

"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", ...
Bib-lost's user avatar
  • 3,900
7 votes
2 answers
1k views

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 ...
freind's user avatar
  • 71
6 votes
2 answers
219 views

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 ...
Hyperplane's user avatar
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4 votes
2 answers
2k views

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 $$\mathcal F_a=\{A\subseteq X; a\in A\}$$ ...
fatemeh's user avatar
  • 377
4 votes
3 answers
2k views

Meaning behind Filter in Set Theory

In a course in logic and set theory, we studied the concept of a Filter. We defined a filter $F \in P(S)$ on $S$ an equivalent of the following definition from Jech's Introduction to Set Theory: (a) $...
ikoikoia's user avatar
  • 329
4 votes
1 answer
240 views

Question about definition of $\kappa$-completeness of filter

I am looking at the following definition: Let $\kappa$ be a regular uncountable cardinal and let $\mathcal F$ be a filter on a non-empty set $X$. We say that $\mathcal F$ is $\kappa$-complete if $\...
Rudy the Reindeer's user avatar
4 votes
1 answer
359 views

Neighbourhood filter of a uniform space

The Wikipedia page for Uniform space includes the following section: Every uniform space $V$ becomes a topological space by defining a subset $O$ of $X$ to be open if and only if for every $x$ in $...
semisilent's user avatar
2 votes
1 answer
298 views

Equivalent characterizations of ultrafilters

If $\mathcal{F}$ is a filter on $X$, will the below conditions be equivalent? (1) $\mathcal{F}$ is an ultrafilter. (2) For every $ \emptyset \neq M \subset X$, either $M \in \mathcal{F}$ or $X ...
fatemeh's user avatar
  • 377
2 votes
1 answer
304 views

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 :...
Maxime Ramzi's user avatar
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2 votes
1 answer
417 views

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 ...
ℋolo's user avatar
  • 10k
1 vote
1 answer
121 views

Question about Stone-Čech compactification

Assume $R=(\mathcal{U}_n)_n$ is a sequence of distinct ultrafilters on some set $X$. Since every Hausdorff space has an infinite discrete subspace, there is a subsequence $R=(\mathcal{V}_n)_n$ of $R=(\...
Ebi's user avatar
  • 185
1 vote
1 answer
121 views

Every minimal KC space is compact

A space $(X,\tau )$ is said to be minimal $KC$ if $(X,\tau)$ is $KC$ but no topology on $X$ which is strictly smaller than $\tau$ is $KC$. Theorem : Every minimal KC space is compact. Proof. ...
fatemeh's user avatar
  • 55
1 vote
1 answer
625 views

Non-principal ultrafilter containing a finite set

Let $I$ be an infinite set. Is there a non-principal ultrafilter $F$ over $I$ which contains a finite subset of $I$?
user695172's user avatar
0 votes
1 answer
542 views

Alexandroff compactification: continuous function extension

Let $(X, \mathcal{T})$ be a non compact topological space, $\infty \notin X$ and $(X^* := X \cup \{\infty\}, \mathcal{T}^* := \{U \subseteq X^*\mid U \cap X \in \mathcal{T} \land (\infty \in U \...
user avatar
27 votes
4 answers
6k views

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 ...
Canis Lupus's user avatar
  • 2,565
27 votes
0 answers
664 views

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 ...
Jakub Konieczny's user avatar
18 votes
0 answers
311 views

Non-rigid ultrapowers in $\mathsf{ZFC}$

This question stems from a weakness in this recent answer of mine. Question: Can $\mathsf{ZFC}$ prove, without invoking first-order model theory, that for every countably infinite structure $\mathcal{...
Noah Schweber's user avatar
17 votes
1 answer
643 views

What makes Fubini product (tensor product) of filters a natural operation?

In several texts you can encounter tensor product or Fubini product of two filters defined as follows: If $p$ is a filter on a set $X$, $q$ be a filter on a set $Y$, then $$p\otimes q=\{A\subseteq X\...
Martin Sleziak's user avatar
17 votes
3 answers
694 views

Generalization of $f(\overline{S}) \subset \overline{f(S)} \iff f$ continuous

A common characterization of a continuous function $f: X \to Y$ is that for any $S \subset X$, $f(\overline{S}) \subset \overline{f(S)}$. Similarly, closed maps are such that $f(\overline{S}) \supset \...
Eric Auld's user avatar
  • 28.2k
14 votes
1 answer
241 views

Nonisomorphic free ultrafilters on $\omega$

Any bijection from $\Bbb N$ to itself transforms an ultrafilter on $\Bbb N$ to another (isomorphic) ultrafilter. Any two principal ultrafilters are isomorphic in that sense. For free ultrafilters on $...
PatrickR's user avatar
  • 4,292
13 votes
3 answers
2k views

Why were filters and nets in topology named filters and nets?

I am wondering why do mathematicians categorizes some structures and called them filters , Nets? In English, filter means: A porous material through which a liquid or gas is passed in order to ...
Hassan Muhammad's user avatar
12 votes
1 answer
575 views

Injective function and ultrafilters

An exercise left by my teacher let me think that the following statement is true: Let $\mathcal{U}$ be an ultrafilter on $\mathbb{N}$. Then every injective function $g:\mathbb{N}\rightarrow\mathbb{...
Jacob Fox's user avatar
  • 1,705
10 votes
2 answers
231 views

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 ...
goblin GONE's user avatar
  • 67.8k
10 votes
1 answer
455 views

How many ultrafilters there are in an infinite space?

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 $X$. ...
Carlos Jiménez's user avatar