Filters and Nets

Some basic examples

Motivation

\[ \gdef\vois#1#2{\mathcal{V}_{#1}(#2)} \]

Nets and filters are used for describing convergence in a non-metric space $X$.

Denote the collection of (open) neighbourhoods of $x \in X$ by \(\vois{X}{x}\).

Definitions and examples

Directed set : A partially ordered set $I$ such that \(\forall i, j \in I: i \le j, \exists k \in I: k \ge j.\)

Net : A function in $X^I$, where $I$ is a directed set.

- example: any sequence in $X^\N$

Convergence of nets to a point : $x_i \to x$ if \(\forall A \in \vois{X}{x}, \exists j \in I: \forall k \ge j, x_k \in A.\)

- example: absolute convergence of series ($I$ is the collection of finite
subsets of $\N$, finite sum $\Sigma \in \R^I$.)
- example: Riemann integral ($I$ is the collection of tagged partitions,
the partial order _doesn't_ depend on tags, $\int \in \R^I$.)

Filter base : A nonempty collection $\mathcal{F} \subseteq \mathcal{P}(X) \setminus {\varnothing}$ such that \(\forall F,G \in \mathcal{F},\exists H \in \mathcal{F}: H \subseteq F \cap G.\) (contains nonempty part of intersection)
Difference with topological basis: sets have to be nonempty here

Filter : A filter base $\mathcal{F}$ so that

- contains supersets: $\forall F \in \mathcal{F}, \forall G \supseteq F, G \in \mathcal{F}$
- contains intersection: $\forall F, G \in \mathcal {F}, F \cap G \in \mathcal{F}$

The image of a filter $\mathcal{F}$ under a function $f$ is also a filter, denoted by $f[[\mathcal{F}]]$.

Filters can be made from a filter base.

Let $\mathcal{F}$ be a filter base. Then $\mathcal{G} = \lbrace G \subseteq X \mid \exists F \in \mathcal{F}: F \subseteq G\rbrace$ is a filter. (We say $\mathcal{F}$ is a filter of $\mathcal{G}$.)

A picture guides through the direct verification. Note that all intersections should be nonempty in this context.

Convergence of a filter base to a point : $\mathcal{F} \to x$ if \(\vois{X}{x} \subseteq \mathcal{G}\), where $\mathcal{G}$ is the filter generated by $\mathcal{F}$.

Equivalent ways to define continuity

  1. Open preimage (usual way)
  2. $x_i \to x \Longrightarrow f(x_i) \to x$
  3. $\mathcal{F} \to x \Longrightarrow f[[\mathcal{F}]] \to f(x)$

The axiom of choice is needed to some stages.


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