# Questions tagged [ultrafilters]

The ultrafilters tag has no usage guidance.

217
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### Points in the Stone Cech compactification are intersection of open sets

Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...

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### Is the Tukey order well-founded

Consider the Tukey order restricted to directed orders of the form $(U,\supseteq)$, where $U$ is an ultrafilter on $\omega$. It is defined as follows:
For two ultrafilters $U,W$ on $\omega$, we say ...

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### Maximal intersecting families on $\omega$ that are not ultrafilters

A family ${\cal S}\subseteq{\cal P}(\omega)$ is intersecting if any two members of ${\cal S}$ have non-empty intersection. Zorn's Lemma implies that every intersecting family is contained in a maximal ...

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### Are sequences going to +infty along a ultrafilter on $\omega$ essentialy increasing?

The question is essentially in the title: suppose you have a non-principal ultrafilter $p$ on $\omega$ and a sequence $(u_n)_{n\in \omega}$ of elements of $\omega$ such that the $p$-limit of $(u_n)$ ...

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### Ultralimit of $w^*$-continuous maps

Let $\omega$ be a free ultrafilter on $\mathbb N.$ Let $(\mathcal M_n)$ be a sequence of finite von Neumann algebras. Let $\mathcal N$ be another finite von Neumann algebra and we have maps $\phi_n:\...

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### A question about the Stone-Čech compactification and ultrafilter

Let $X$ be a Tychonoff space and let $\beta X$ is the Stone-Čech
compactification of $X$. Assume $f:X\longrightarrow \mathbb{R}$ is a bounded function. Then there exists a function $f^{\beta }:\beta X\...

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### Mysior plane is not realcompact

Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup ...

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### Must an ultrafilter orthogonal to all ultrafilters containing an element $a$ contains $a^\perp$?

Let $L$ be an orthocomplemented lattice. We may consider the collection $U$ of ultrafilters on $L$. We say two elements $a, b \in L$ are orthogonal to each other, written as $a \perp b$, if $a \leq b^\...

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### The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence

Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are ...

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### Can there be a p-point ultrafilter that is 'aggressively non-Ramsey'?

These are fairly standard terms, but for the sake of completeness: An ultrafilter $\mathcal{U}$ on $\omega$ is a p-point if whenever $(A_n)_{n<\omega}$ is a partition of $\omega$ such that $A_n \...

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### Infinite tensor/Fubini product of ultrafilters

Given an infinite family $\{\mathcal{F}_{\lambda}$, $\lambda <\kappa\}$, $\kappa \geq \omega_0$, of (ultra)filters of a set $X$, how it is defined the infinite tensor/Fubini product $$\bigotimes_{\...

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### Comparing Mathias forcing notions relative to various filters

Let $\mathcal F$ be a (non-principle, non trivial, ...) filter on $\omega$. The Mathias Forcing relative to $\mathcal F$ is the forcing notion $\mathbb M(\mathcal F)$ consisting of pairs $(s, X)$ with ...

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### Ultrafilter lemma for arbitrary lattice

Can someone kindly confirm whether the ultrafilter lemma for arbitrary (i.e., not necessarily Boolean) bounded lattices is equivalent to Zorn's lemma?
To be precise, if $\mathbf{L} = (L, \leq, \land, \...

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### Supremum of infimum of measure of members of a free ultrafilter

For a set $A\subseteq \omega$ we let the upper density of $A$ be defined as $d^+(A) := \lim\sup_{n\to\infty}\frac{|A\cap(n+1)|}{n+1}$. Let $\text{FrU}(\omega)$ be the collection of free ultrafilters ...

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### An ultrafilter on $\omega_1$ with a nice Fubini product with an ultrafilter on $\omega$

Fix an ultrafilter $U$ on $\omega$ (that is, $U$ is an ultrafilter on the Boolean algebra of all subsets of $\omega$).
Let $(f_\alpha \mid \alpha < \omega_1)$ be an increasing sequence in $\mathbb{...

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### Decomposition of an ultrafilter on the fibers of a map

Short version: If I have a map $f:Y \to I$, and $\mu$ an ultrafilter on $Y$, under what condition can $\mu$ be written as a limit/sum/integral of ultrafilters on the fibers of $f$ along the ...

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### A p-point game with infinitely many ultrafilters

The following game-theoretic characterization of p-points is well known:
Theorem A. An ultrafilter $D$ on the set $\omega$ of natural numbers is a p-point if and only if player I does not have a ...

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### Ultrafilter subtraction and "zero"

This is related to a couple recent MO/MSE questions of mine, namely 1,2. Belatedly, I've tweaked this post to remove an overly-ambitious secondary question; see the edit history if interested.
Let $\...

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### ultrapower(ultrapower)=ultrapower

Is there a nonprincipal ultrafilter $\omega$ on $\mathbb N$ such that for any metric space $M$ there is an isometry
$$(M^\omega)^\omega\to M^\omega?$$
(In other words, the $\omega$-power of $\omega$-...

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### Ultrafilters of closed sets

The following definition should be standard, but let me state it just in case there is some ambiguity:
If $\mathscr{L}$ is a set of subsets of a set $X$ that is closed under finite unions and ...

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### Silver-like forcing preserves p-points (reference request)

A Silver forcing "below $2^n$" is defined e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It ...

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### Is $(\omega+1)^\omega/{\cal U}$ complete for ${\cal U}$ free ultrafilter?

Let ${\cal U}$ be a free ultrafilter on $\omega$. Is the linearly ordered set $(\omega+1)^\omega/{\cal U}$ complete?

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### Are all free ultrafilters 'the same' in some sense?

Consider the set of ultrafilters $\beta(\mathbb N)$ on $\mathbb N$.
Any function $f\colon\mathbb N\to\mathbb N$ extends to a function $\beta f\colon \beta \mathbb N \to \beta\mathbb N$. We say that ...

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### Can there be no "surprisingly averageable" second-order sentences?

Say that a second-order sentence $\varphi$ is averageable iff there exists some infinite cardinal $\kappa$ and some nonprincipal ultrafilter $\mathcal{U}$ on $\kappa$ such that for every $\kappa$-...

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

Originally asked and bountied at MSE:
Question: Can $\mathsf{ZFC}$ prove that for every countably infinite structure $\mathcal{A}$ in a countable language there is an ultrafilter $\mathcal{U}$ on $\...

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### Are these two definitions of $\mathcal{U}$-Ramsey set equivalent?

Let $\mathcal{U}$ be an ultrafilter over $\omega$, and let $\mathcal{X} \subseteq [\omega]^\omega$. In two separate texts, there are two possible interpretations of a $\mathcal{U}$-Ramsey set, as ...

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### Ramsey ultrafilters on partial order

$\newcommand{\U}{\mathcal{U}}$
$\newcommand{\P}{\mathbb{P}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\F}{\mathcal{F}}$
Recall the following equivalent definitions of a Ramsey ultrafilter over $\...

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### Is the set of $\kappa$-complete ultrafilters closed in $\beta X$?

Given an arbitrary set $X$, let $\beta X$ be the set of all ultrafilters over $X$. Consider endowing $\beta X$ with a topology consisting of the following open sets:
$$
\{\mathcal{U} \in \beta X : A \...

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### Can Tychonoffs theorem for a countable number of spaces be proven with ZF plus the axiom of (countable) dependent choice?

It can be proven without any form of infinite choice that the product of two compact spaces (and thus any finite product) is compact, while on the other hand, it is well known that the general form of ...

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### On a completeness property of hyperreals

Let $\mathbb{R}_*=\mathbb{R}^\omega/\mathcal U$ for some ultrafilter $\cal U$. In the definitions of this question and assuming ZFC + CH there are only three types of cuts in $\mathbb{R}_*$: $(\omega,\...

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### SPOT as a conservative extension of Zermelo–Fraenkel

In Infinitesimal analysis without the Axiom of Choice, Hrbacek and Katz have shown that it is possible to formulate an axiomatic theory which provides a formalisation of calculus procedures which make ...

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### "Good limit" of an uncountable sequence of elements of an ultrafilter

Let $U$ be an ultrafilter on $\mathcal{P}(\omega)$ and $\langle \sigma _\alpha \mid \alpha < \omega_1 \rangle$ be a sequence of elements of $U$.
I know that the limit sup of $\sigma _\alpha$'s ($= \...

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### Grigorieff forcing and destruction of ultrafilters

I was interested in the Grigorieff forcing (you can read the definition here: How "much" does (Grigorieff) forcing destroy an ultrafilter?)
I couldn't prove that it destroys ultrafilters, ...

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### ${\frak b}$ and ${\frak d}$ in the Rudin-Keisler preordering

If $(Q,\leq)$ is any preordered set (that is, $\leq$ is a reflexive and transitive, but not necessarily anti-symmetric relation), then we say that $S\subseteq Q$ is
unbounded if for all $q\in Q$ ...

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### Is the Rudin-Keisler ordering a continuous relation?

If $X, Y$ are topological, and $R\subseteq X\times Y$ we say that $R$ is continuous (from $X$ to $Y$) if for every $V\subseteq Y$ with $V$ open, we have $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\...

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### "Completion property" in $(\beta\omega,+)$

Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. For ${\bf a}, {\bf b}\in \beta\omega$, ...

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### Minimal components of the translation action on the Stone–Čech compactification

$\newcommand\Cb{C^\text b}$Let $\Cb(\mathbb R)$ be the C*-algebra formed by all bounded, continuous, complex valued functions on $\mathbb R$.
Consider the action $\tau $ of $\mathbb R$ on $\Cb(\...

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### Addition and Rudin-Keisler ordering in $\beta \omega$

$\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that extends ...

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### Finite pre-orders embeddable in the Rudin-Keisler ordering

$\DeclareMathOperator{\NPU}{\operatorname{NPU}}\DeclareMathOperator{\RK}{\,\mathrm{RK}}$A pre-ordered set is a pair $(P, \leq)$ where $P$ is a set and $\leq\subseteq P\times P$ is a reflexive and ...

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### Topological complexity of ultrafilters in $2^\kappa$ for uncountable $\kappa$

It is a well known fact that if $\mathcal{F}$ is a non-principal ultrafilter on $\omega$, then the set $\{ \alpha \in 2^\omega : \alpha \in \mathcal{F}\}$ (conflating binary strings with subsets of $\...

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### Unbounded $\omega_1$-sequence in $^*\mathbb{N}$

Let $\mathcal{F}$ be a non-principal ultrafilter on $\omega$. Let $^*\mathbb{N}$ = $\mathbb{N}^\omega/\mathcal{F}$ be an ultrapower. Let $\{n_\alpha\}_{\alpha\in\omega_1}$ be a strictly increasing ...

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### Regularity and ultrafilter

I read the following result in an article.
Let $X$ be a regular space. Let $\mathcal{M}$ be free closed
ultrafilter on $X$. Set $\mathcal{U=}\left\{ U:U\text{ is open and there
exists a }F\in \mathcal{...

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### When do two ultrafilters yield isomorphic ultrapowers?

Fix a cardinal $\lambda$$\newcommand{\cU}{\mathcal U}\newcommand{\cV}{\mathcal V}$. Consider the equivalence relation on $\beta\lambda$ given by $\cU\sim \cV$ when for all first-order structures $M$ ...

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### External ultrafilters definitions

I am reading a paper by Goldberg, in which he defines ultrafilter over a model of set theory (not transitive). These are the definitions:
I get the definition of an M-ultrafilter, it is a real subset ...

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### Multiplicative and additive groups of the field $(\prod_{n\in\omega}\mathbb{Z}/p_n\mathbb{Z})/\simeq_{\cal U}$

Let ${\cal U}$ be a non-principal ultrafilter on $\omega$, and for each $n\in\omega$, let $p_n$ denote the $n$th prime, that is $p_0 = 2, p_1=3, \ldots$
Next we introduce the following standard ...

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### Ultralimit of metric spaces vs. inductive limits of underlying topological spaces

Let $\{(X_n,d_n)\}_{n =1}^{\infty}$ be a sequence of bounded metric spaces such that:
$X_n \subseteq X_{n+1}$ is a metric subspace of $X_n$.
Let $\omega$ denote a non-principal ultrafilter (i.e.: ...

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### Regular limit points of possible cofinalities

Let $A$ be a non-empty set of regular cardinals such that $\vert A\vert <\text{min}\ A$, and $\{\nu_i\mid i<i_0\}\subseteq \text{pcf}\ A$ be a strict increasing sequence having limit length $i_0$...

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### What is the smallest cardinality of a base of an ultrafilter on $\omega$ related to an almost disjoint family of cardinality $\mathfrak c$?

Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an almost disjoint family of infinite subsets of $\omega$. The almost disjointness of the family means that $A_\alpha\cap A_\beta$ is finite for any ordinals ...

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### Notation of $P^+$-families - bibliography searching

have you ever met with notation of $P^+$-families in other papers than Iian B. Smythe "A local Ramsey theory for block sequences" and his phd?
Thank you in advance

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### NCF, P-points, weak P-points, and cardinalities

The post is a bit long, but all the questions are similar or concern the same topic.
Let $\omega^*=\beta\omega\setminus\omega$. A well-known topological definition of a P-point (on $\omega$) is as ...