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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27 votes
4 answers
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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 ...
Canis Lupus's user avatar
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27 votes
0 answers
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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 ...
Jakub Konieczny's user avatar
25 votes
1 answer
964 views

Showing a filter on the Power set of $\mathbb{Z}$ is a one point Filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in X^{\...
Dominic Michaelis's user avatar
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
20 votes
5 answers
4k views

Introduction to Filters in Topology

Question: What are some good resources for a student who has taken algebraic and point-set topology and who wishes to learn how filters and ultrafilters are applied in topology? Motivation: I've seen ...
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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19 votes
3 answers
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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 ...
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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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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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
3 answers
693 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
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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
16 votes
2 answers
3k views

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
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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14 votes
3 answers
1k views

Can you get a non-principal ultrafilter on N using Choice but 'avoiding' Zorn's Lemma?

This question is not about the logical relationships between Choice, Zorn's and the Ultrafilter Lemma, but pedagogical. I am teaching a class and want to go as directly as possible from Choice to a ...
Thompson's user avatar
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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
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14 votes
1 answer
2k views

Every net has an ultranet as subnet: direct proof

I'm currently brushing up my topology using Willard's General Topology. Currently I'm working through the chapters 11 and 12 on nets and filters. Chapter 12 deals extensively with ultrafilters and ...
Lord_Farin's user avatar
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13 votes
3 answers
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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
13 votes
1 answer
753 views

Cauchy filters in metric spaces

Some terminology: Let $(X,d)$ be a metric space. A filter $\mathcal F \subseteq \mathcal P (X)$ is Cauchy if $\forall \epsilon >0 \exists x\in X: B_\epsilon(x)\in \mathcal F$. A filter $\mathcal ...
Ittay Weiss'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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12 votes
1 answer
6k views

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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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
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11 votes
0 answers
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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 ...
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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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
  • 133
10 votes
2 answers
1k views

Intuition for the Stone-Čech compactification via ultrafilters

Definitions used: Given some set $X$, denote by $\beta X$ the set of ultrafilters on $X$. We can view $X$ as a subset of $\beta X$ by identifying each point $x \in X$ with the principal ultrafilter ...
Jack's user avatar
  • 155
10 votes
2 answers
631 views

A Prime $\mathcal P$-filter is contained in a unique $\mathcal P$-ultrafilter?

Some backround: Let $\mathcal P$ be a class of subsets of a topological space such that if $P_1$ and $P_2$ are sets from $\mathcal P$ then $P_1\cap P_2$ and $P_1\cup P_2$ belong to $\mathcal P$. A $\...
Camilo Arosemena-Serrato's user avatar
10 votes
1 answer
174 views

Is every "filter" of rings principal?

Let $\mathcal{F}$ be a class of rings* with the following property: If there is a homomorphism of rings $A \to B$ with $A \in \mathcal{F}$, then $B \in \mathcal{F}$. Moreover, $\mathcal{F}$ is closed ...
Martin Brandenburg'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
10 votes
1 answer
257 views

Hausdorff spaces from filters

I'm sure I'm just being silly, but I've run into a claim in a paper I'm reading which I don't understand. Suppose $\mathcal{F}$ is a filter on $\mathbb{N}$. There is a natural topology $\tau_\mathcal{...
Noah Schweber's user avatar
10 votes
1 answer
426 views

If $\kappa$ is measurable, does there exist a normal measure on $\mathcal P_{\kappa}(\kappa)$?

I'm trying to do exercise $10.7$ of Jech's Set Theory: If $\kappa$ is a measurable cardinal, then there exists a normal measure on $\mathcal P_{\kappa}(\kappa)$. For a set $A$, with $|A|\geq \...
Camilo Arosemena-Serrato's user avatar
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
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
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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
262 views

Two questions on completely regular filters in locales

I'm reading the exposition of the Stone-Čech compactification for locales in Johnstone's book Stone Spaces. In Chapter IV Paragraph 2.2, Johnstone constructs the Stone-Čech compactification of a ...
Alex Kruckman's user avatar
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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9 votes
1 answer
507 views

Non-Isomorphic Ultrapowers

It is clear that given a family $(\mathfrak{A}_i)_{i\in I}$ of $L$-structures their ultraproduct may depend on the choice of the ultrafilter (for this question I am only considering non-principal ...
Santiago C.'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
1 answer
920 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
Jakub Konieczny's user avatar
9 votes
2 answers
370 views

Proof of non-existence of non-principal $\kappa$-complete ultrafilter

Let $\lambda$ be a cardinal. I would like to prove that for all cardinals $\lambda < \kappa \leq 2^\lambda$, there can't be a $\kappa$-complete non-principal ultrafilter on $\kappa$. Here is my ...
zarathustra's user avatar
  • 4,901
9 votes
2 answers
105 views

On $\mathcal{A}$ with $\forall C\in\mathcal{P}_{\infty}(\mathbb{N})\exists A\in\mathcal{A}:A\cap C\in\mathcal{P}_{\infty}(\mathbb{N})$

Edit: I came up with an answer myself, see below. Denote $\mathcal{P}_{\infty}(\mathbb{N})=\{A\subseteq\mathbb{N}\ |\ A\text{ infinite}\}$ and let $\mathcal{A}\subseteq \mathcal{P}_{\infty}(\mathbb{N}...
Redundant Aunt's user avatar
9 votes
1 answer
316 views

Can ultrapowering add choice?

I'm supervising a reading course in set theory, and the following question came up (let $\mathsf{LT}$ be Łoś's Theorem; in an earlier version of this question I mistakenly thought $\mathsf{LT}$ was ...
Noah Schweber's user avatar
8 votes
2 answers
4k views

Non-Principal Ultrafilters Confused!!

I've just started learning about filters and non-principal ultrafilters. I'm getting confused on the requirement: $U$ contains no finite subsets of $J$; where $U$ is the ultrafilter and $J$ is a set. ...
user avatar
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
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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8 votes
4 answers
427 views

Two topologies are equal if they have the same filter convergence

A major drawback with sequential convergence in topological spaces is that two different topologies can have the same convergent sequences e.g. the discrete and cofinite topologies on $\mathbb{R}$. ...
Gr Eg's user avatar
  • 239
8 votes
1 answer
721 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
2 answers
824 views

Intuition about filters

A filter is defined as follows: Let $X$ be a non-empty set. A non-empty family $\mathcal{F}\subset \mathcal{P}(X)$ is a filter on $X$ if: $A\neq \emptyset$ for all $A\in \mathcal{F}$, If $A,B\in \...
Gold'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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