# Tagged Questions

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.

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### Number of equivalence classes based on a relation regarding a non-principal ultrafilter

We have an equivalence relation on $\mathbb{N}^\mathbb{N}$ given by $$f\equiv g \iff \{n\in\mathbb{N}: f(n)=g(n)\}\in\mathbb{U},$$ where $\mathbb{U}$ is a non-principal ultrafilter on $\mathbb{N}$. ...
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### Proof of Ultrafilter lemma with two propositions and Zorn lemma

I would like to prove the following: Let $X$ be any set, then every filter $\mathcal{F}$ on $X$ is contained in an ultrafilter $F$ Using two propositions and Zorn Lemma. I am required to come ...
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### Can someone please offer a simple definition of “derived net”?

I was looking up a term called "derived net", however the Google search seems to conflate "net" with .NET programming language. (And filter with electronic filters, and "derived net from a filter" ...
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### Show that if $x$ is an accumulation point of an ultrafilter on $X$, then the neighborhood filter is contained in the filter

Show that if $x$ is an accumulation point of an ultrafilter $\mathcal{F}$ on $X$, then the neighborhood filter $\mathcal{F}_x$ is contained in the filter i.e. $\mathcal{F}_x \subseteq \mathcal{F}$ ...
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### Any subset of elements in an ultrafilter is in the ultrafilter

I am trying to prove the following fact: Given $\mathcal{F}$ an ultrafilter on $X$, suppose $A \in \mathcal{F}$ and $B \subseteq A$, then either $B$ or $A\backslash B$ are in $\mathcal{F}$ ...
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### Given $\mathcal{F}$ a filter on $X$, and either $A$ or its complement are in $\mathcal{F}$, then $\mathcal{F}$ is ultrafilter

I am trying to prove the following fact: Given $\mathcal{F}$ a filter on $X$, Suppose $\forall A \subseteq X$ where either $A$ or $X\backslash A$ are in $\mathcal{F}$, then $\mathcal{F}$ is ...
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### Product of Ultrafilters: How to show $p,q <_{RK} p\otimes q$?

Let $p,q$ be two free ultrafilter on $\mathbb{N}$, i.e. elements of $\mathbb{N}^* = \beta\mathbb{N}\setminus\mathbb{N}$. The Rudin-Keisler order is defined as follows: $p \leq_{RK} q$ iff there is a ...
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### Ultrafilter on $[0,1]$ consisting of closed sets

Today we learned about filters and ultrafilters in the General Topology course. I am trying to play around with these definitions. I wish to ask a question that I am unsure about. Let us say, we have ...
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### Why are the ultrafilters on $\mathbb{N}$ precisely the minimal Cauchyfilters?

One can place an equivalence relation on the set of all Cauchyfilters in a uniform space: $\mathcal{U}\sim\mathcal{V}$ iff $\mathcal{U}\cap\mathcal{V}$ is a Cauchyfilter. In my topology course, we ...
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### Why do ultraproduct structures use a quotient as their universe?

For an $L$-structure $\mathfrak{A}$ with universe $A$, if we have an index set $I$, with an ultrafilter $U$, we create an ultraproduct structure having as its universe $\Pi_I \;A_i/U$. This is the set ...