Nets and Filters

In metric spaces, sequences suffice to characterize the topology: closures, continuity, and compactness can all be described via convergent sequences. In general topological spaces, sequences are inadequate, and two more powerful tools -- nets and filters -- take their place.

Directed Sets and Nets

Definition8.1Directed Set

A directed set $(I, \preceq)$ is a nonempty set $I$ with a preorder $\preceq$ (reflexive and transitive) such that for every $\alpha, \beta \in I$ , there exists $\gamma \in I$ with $\alpha \preceq \gamma$ and $\beta \preceq \gamma$ .

Definition8.2Net

A net in a topological space $X$ is a function $x: I \to X$ from a directed set $I$ to $X$ , written $(x_\alpha)_{\alpha \in I}$ .

A net $(x_\alpha)$ converges to $x \in X$ , written $x_\alpha \to x$ , if for every open neighborhood $U$ of $x$ , there exists $\alpha_0 \in I$ such that $x_\alpha \in U$ for all $\alpha \succeq \alpha_0$ .

ExampleExamples of Nets

Sequences: A sequence is a net indexed by $(\mathbb{N}, \leq)$ .
Neighborhood nets: For $x \in X$ , let $I = \mathcal{N}(x)$ (the neighborhood system of $x$ ) directed by reverse inclusion: $U \preceq V$ iff $U \supseteq V$ . For each $U \in I$ , choose $x_U \in U$ . Then $(x_U)$ is a net converging to $x$ .
Riemann sums: The net of Riemann sums of a function $f$ on $[a, b]$ , indexed by partitions ordered by refinement, converges to $\int_a^b f$ (when the integral exists).

Nets Characterize the Topology

Theorem8.1Nets Characterize Closure and Continuity

Let $X$ be a topological space.

Closure: $x \in \overline{A}$ if and only if there exists a net $(x_\alpha)$ in $A$ converging to $x$ .
Continuity: $f: X \to Y$ is continuous if and only if for every net $(x_\alpha)$ converging to $x$ in $X$ , the net $(f(x_\alpha))$ converges to $f(x)$ in $Y$ .
Hausdorff: $X$ is Hausdorff if and only if every net in $X$ converges to at most one point.

Proof

(1, $\Leftarrow$ ): If $(x_\alpha) \to x$ with $x_\alpha \in A$ , every open $U \ni x$ contains some $x_\alpha \in A$ , so $x \in \overline{A}$ .

(1, $\Rightarrow$ ): If $x \in \overline{A}$ , let $I$ be the collection of open neighborhoods of $x$ , directed by reverse inclusion. For each $U \in I$ , $U \cap A \neq \emptyset$ ; choose $x_U \in U \cap A$ . Then $(x_U)$ is a net in $A$ converging to $x$ .

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Filters

Definition8.3Filter

A filter on a set $X$ is a nonempty collection $\mathcal{F} \subseteq \mathcal{P}(X)$ satisfying:

$\emptyset \notin \mathcal{F}$ .
If $A, B \in \mathcal{F}$ , then $A \cap B \in \mathcal{F}$ (closed under finite intersections).
If $A \in \mathcal{F}$ and $A \subseteq B \subseteq X$ , then $B \in \mathcal{F}$ (upward closed).

Definition8.4Filter Convergence

A filter $\mathcal{F}$ on a topological space $X$ converges to $x$ , written $\mathcal{F} \to x$ , if $\mathcal{F}$ contains all neighborhoods of $x$ : $\mathcal{N}(x) \subseteq \mathcal{F}$ .

A filter $\mathcal{F}$ clusters at $x$ (or $x$ is a cluster point of $\mathcal{F}$ ) if every $F \in \mathcal{F}$ intersects every neighborhood of $x$ .

ExampleExamples of Filters

Neighborhood filter: $\mathcal{N}(x) = \{N \subseteq X : x \in \operatorname{int}(N)\}$ is the smallest filter converging to $x$ .
Principal filter: For nonempty $A \subseteq X$ , $\mathcal{F}_A = \{B \subseteq X : A \subseteq B\}$ is a filter (the principal filter generated by $A$ ).
Fr'echet filter: On an infinite set $X$ , the collection of cofinite subsets $\{A \subseteq X : X \setminus A \text{ is finite}\}$ is a filter.

Filter Base

Definition8.5Filter Base

A filter base (or filter basis) on $X$ is a nonempty collection $\mathcal{B} \subseteq \mathcal{P}(X)$ such that:

$\emptyset \notin \mathcal{B}$ .
For $B_1, B_2 \in \mathcal{B}$ , there exists $B_3 \in \mathcal{B}$ with $B_3 \subseteq B_1 \cap B_2$ .

A filter base $\mathcal{B}$ generates a filter: $\langle \mathcal{B} \rangle = \{A \subseteq X : \exists B \in \mathcal{B}, B \subseteq A\}.$

RemarkNets vs. Filters

Nets and filters provide equivalent descriptions of topology. Every net generates a filter (the eventuality filter: $A \in \mathcal{F}$ iff $x_\alpha \in A$ eventually). Conversely, every filter generates a net. The choice between nets and filters is largely a matter of taste:

Nets are more intuitive (they generalize sequences directly).
Filters are more algebraic and have cleaner behavior under products and intersections.
Ultrafilters (maximal filters) are the key tool for proving Tychonoff's theorem.