### 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

- Open preimage (usual way)
- $x_i \to x \Longrightarrow f(x_i) \to x$
- $\mathcal{F} \to x \Longrightarrow f[[\mathcal{F}]] \to f(x)$

The axiom of choice is needed to some stages.