# 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.

941 questions
Filter by
Sorted by
Tagged with
4k views

### 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 ...
• 4,041
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 ...
• 2,565
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 ...
• 12.6k
964 views

• 53.8k
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 ...
• 53.8k
1k views

• 4,292
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 ...
• 17.8k
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 ...
• 4,292
753 views

• 1,705
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 ...
• 53.8k
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
• 22.6k
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 ...
• 133
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 ...
• 155
631 views

• 246k
426 views

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 $|A|\geq \... 10 votes 1 answer 455 views ### How many ultrafilters there are in an infinite space? I'm stuck with the next exercise from the book Rings of Continuous Functions by Gillman. If$X$is infinite, there exist$2^{2^{|X|}}$ultrafilters on$X$all of whose members are of cardinal$X$. ... • 3,895 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 ... • 67.8k 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 ... • 1,949 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 ... • 76.6k 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. • 881 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 ... • 884 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 ... • 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 ... • 12.6k 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 ... • 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}...
316 views

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 ...
• 246k
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. ...
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 ...
• 87.5k
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 ...
• 47.4k
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 $\mathbb{R}$. ...
• 239
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 ...
• 2,865