Review of Filters and Ultrafilters
Review of Filters and Ultrafilters
- Recall from the Filters and Filter Bases page that if $E$ is a set then a Filter is a collection $\mathscr{F}$ of nonempty subsets of $E$ with the properties that if $A, B \in \mathscr{F}$ then $A \cap B \in \mathscr{F}$, and that if $A \in \mathscr{F}$ and $A \subseteq B$ then $B \in \mathscr{F}$.
- We noted that if $E$ is a set and $A \subseteq E$ then the collection of all subsets of $E$ which contain $A$ is a filter. Furthermore, if $E$ is a topological space and $a \in E$ then the collection of all neighbourhoods of $a$ is a filter.
- We said that a Refinement of $\mathscr{F}$ is a filter $\mathscr{G}$ such that $\mathscr{F} \subseteq \mathscr{G}$.
- We then defined a Filter Base to be a nonempty collection $\mathscr{B}$ of nonempty subsets of $E$ with the property that if $A, B \in \mathscr{B}$ then there exists a $C \in \mathscr{B}$ such that $C \subseteq A \cap B$.
- The Filter Generated by the Filter Base $\mathscr{B}$ is the collection of all subsets which contain a set in $\mathscr{B}$, which we verified was indeed a filter.
- We proved that if $E$ and $F$ are sets and $f : E \to F$ a function, then if $\mathscr{B}$ is a filter base in $E$ then $f(\mathscr{B})$ is a filter base in $F$.
- On the Convergence of Filters and Filter Bases in a Topological Space page, for a topological space $E$, we discussed what it means for a filter $\mathscr{F}$ to converge to a point $a$. We summarize this notion below.
Equivalent Statements for Filter Convergence | |
(1) | $\mathscr{F}$ converges to $a$, i.e., every neighbourhood of $a$ contains a set from $\mathscr{F}$. |
(2) | Every neighbourhood of $a$ is in $\mathscr{F}$. |
(3) | $\mathscr{F}$ is a refinement of the filter consisting of the neighbourhoods of $a$. |
- We then proved some useful factors about convergent filters. We noted that:
If $\mathscr{F}$ converges to $a$ then $a$ is contained in the closure of every set in the filter.
- We also noted the uniqueness of limits for convergent filters in a Hausdorff topological space. That is:
If $E$ is a Hausdorff topological space and if $\mathscr{F}$ is a filter that converges to $a$ and $b$ then $a = b$.
- On The Elementary Filter Associated with a Sequence we noted that filters are essentially generalizations of sequences. If $(x_n) \subseteq E$ is a sequence and if for each $n \in \mathbb{N}$ we define:
\begin{align} \quad X_n := \{ x_n : i \geq n \} \end{align}
- then $\{ X_n : n \in \mathbb{N} \}$ is a filter base, and the Elementary Filter Associated to $(x_n)$ is the filter generated by $\{ X_n : n \in \mathbb{N} \}$. We proved the following:
In a topological space $E$, a sequence $(x_n)$ converges to $a$ if and only if the elementary filter associated to $(x_n)$ converges to $a$.
- On the Ultrafilters page we define an Ultrafilter to be a filter with no proper refinement. We summarize some properties of ultrafilters below.
Properties of Ultrafilters | |
(1) | If $\mathscr{F}$ is an ultrafilter and $A \subseteq E$ then either $A \in \mathscr{F}$ or $A^c \in \mathscr{F}$. |
(2) | If $\mathscr{F}$ is an ultrafilter and $\displaystyle{\bigcup_{i=1}^{n} A_i \in \mathscr{F}}$ then $A_i \in \mathscr{F}$ for some $1 \leq i \leq n$. |