# 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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### Does the filter, $F$, on $S$ exist such that $p,q\in S$ and $p,q\in \lim{F}$

Consider two points $p,q\in S$ with $p\ne q$. Is it possible to find a filter, $F$, on $S$ such that all neighborhoods of $p$ and $q$ are contained in $F$? I would assume that when $p\ne q$ then, in ...
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### Measurable cardinals, elementary embeddings, and Kunen's theorem

Suppose $\kappa$ is a measurable cardinal. Then if $U$ is the ultrafilter on $\kappa$, we can use this to generate an ultrapower of the entire universe. We can then embed $V$ into this ultrapower in ...
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### How can we prove that the ultrafilter lemma is not provable from ZF?

The ultrafilter lemma states that every proper filter on a set X is contained in some ultrafilter on X. Wikipedia says that in ZFC one can prove the ultrafilter lemma, but in ZF it's not possible to ...
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### Certain properties of uniform structure

I am trying to read about unifrom spaces from Introduction to Uniform Spaces, and I was wondering about some basic facts which I wasn't able to find there. Let $(X,\mathcal{E})$ be a uniform space. ...
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### How to fuse inertial and optical tracking data with an error-state kalman filter, and provide pose estimations for both update types?

I am searching for an error-state kalman filter that is able to fuse inertial and optical tracking data but provides pose estimates for both optical and inertial updates. Currently I am using the ...
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### ultrafilters as linear orders

In Henson's Model Theory lecture notes I found an exercise quite early on (1.30, p. 12) that prove too difficult for me. It goes like this: Let $L$ be the first order language whose only nonlogical ...
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### Relation between a filter and its complement being an ideal.

Exercise. Show that a true filter $\mathcal{F}$ in $X$ is an ultrafilter iff the complement of $\mathcal{F}$ is an ideal, i.e., $Y,Z \notin \mathcal{F} \Leftrightarrow Y \cup Z \notin \mathcal{F}$, ...
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### Simple proof about prime filters.

Exercise. Let $F$ be an true filter in a set $X$. Show that $F$ is an ultrafilter ($\Leftrightarrow$ prime) iff a subset of $X$ intersects every set of $F$ then this same subset is in $F$. UPDATE. My ...
In this answer, if I understand it right, the following theorem is stated: Let $X$ be a set and $f:X \to X$ an injective map. Let $\mathcal U$ be an ultrafilter on $X$ such that $f(U) \in \mathcal U$ ...
### Prove a map $\varphi$ between all nets and all filters is surjective.
We define such a map: $\varphi(N) = \{A \subset X \ | \ \exists \alpha \in \Omega, \forall \beta \ge \alpha: f(\beta) \in A \}$ Here $N = (\Omega, f, \ge)$ is a net with index set $\Omega$. This way \$\...