In [metric spaces](/page/Metric%20Space), [sequences](/page/Sequence) characterise the [topology](/page/Topology) completely: a set is [closed](/page/Closed%20Set) if and only if it contains the limits of all its convergent sequences, a function is [continuous](/page/Continuous) if and only if it preserves sequential limits, and [compactness](/page/Compact%20Space) is equivalent to sequential compactness. These sequential characterisations are the workhorse of classical analysis, underpinning everything from the Bolzano--Weierstrass theorem to the Arzel\`a--Ascoli theorem.

In general topological spaces, however, sequences fail to detect the topology. The fundamental obstruction is that a topology may have "too many" open sets relative to the countable index set $\mathbb{N}$: a sequence, being a function from $\mathbb{N}$, can only "probe" countably many neighbourhoods, and in spaces that are not [first-countable](/page/First-Countable%20Space), this is not enough to determine convergence. This failure is not a curiosity confined to pathological spaces --- it arises in mainstream mathematics whenever one works with uncountable products, [weak topologies](/page/Weak%20Topology) on non-separable Banach spaces, or spaces of distributions.

[example: Sequential Closure Failure in an Uncountable Product] Consider the product space $X = \{0, 1\}^{\mathbb{R}}$, the set of all functions $f: \mathbb{R} \to \{0, 1\}$ equipped with the [product topology](/page/Product%20Topology). Each factor $\{0, 1\}$ carries the discrete topology. Define the subset \begin{align*} A := \{f \in X : f(t) = 0 \text{ for all but finitely many } t \in \mathbb{R}\}, \end{align*} consisting of functions with finite support. We claim that the constant function $g \equiv 1$ belongs to the [closure](/page/Closure) $\overline{A}$ but is not the limit of any sequence in $A$. **Closure membership.** A basic open set in the product topology has the form \begin{align*} U = \prod_{t \in \mathbb{R}} U_t, \quad \text{where } U_t = \{0, 1\} \text{ for all but finitely many } t. \end{align*} Suppose $g \in U$. Then $U_t = \{1\}$ for only finitely many values of $t$, say $t_1, \ldots, t_k$. The function $f$ defined by $f(t_i) = 1$ for $i = 1, \ldots, k$ and $f(t) = 0$ for all other $t$ belongs to both $A$ and $U$. Hence every basic open neighbourhood of $g$ meets $A$, so $g \in \overline{A}$. **No sequence in $A$ converges to $g$.** Suppose for contradiction that $(f_n)_{n \in \mathbb{N}}$ is a sequence in $A$ with $f_n \to g$ in the product topology. Each $f_n$ has finite support, so the set $S := \bigcup_{n=1}^\infty \operatorname{supp}(f_n)$ is a countable union of finite sets, hence countable. Since $\mathbb{R}$ is uncountable, there exists $t_0 \in \mathbb{R} \setminus S$. For this $t_0$, we have $f_n(t_0) = 0$ for every $n$, so $\pi_{t_0}(f_n) = 0$ for all $n$. But $\pi_{t_0}(g) = 1$, and since the projection $\pi_{t_0}: X \to \{0,1\}$ is continuous, convergence $f_n \to g$ would require $\pi_{t_0}(f_n) \to \pi_{t_0}(g) = 1$, a contradiction. [/example]

This example reveals a genuine gap in the theory: the [closure](/page/Closure) page on Androma notes that in metric spaces, $x \in \overline{A}$ if and only if some sequence in $A$ converges to $x$, but in the product space above, the sequential closure of $A$ (the set of all sequential limits from $A$) is strictly smaller than $\overline{A}$. Sequences cannot reach every point of the closure.

Two solutions to this problem emerged in the 1930s and 1940s. **Nets**, introduced by Moore and Smith (1922) and systematised by Kelley (1955), generalise sequences by replacing the index set $\mathbb{N}$ with an arbitrary directed set --- a partially ordered set in which every finite subset has an upper bound. **Filters**, introduced by Cartan (1937), take a dual approach: instead of tracking a generalised sequence of points, one tracks a family of "large" subsets of the space that are eventually entered. Both frameworks recover the full power of sequential characterisations in arbitrary topological spaces, and they are equivalent in a precise sense: every net determines a filter (its "eventuality filter"), and every filter determines a class of nets (any net whose tails generate the filter).

This page develops both theories in parallel, emphasising the problems each solves and the contexts in which each is most natural.

## Definition

The key idea behind nets is to replace the index set $\mathbb{N}$, ordered by $\leq$, with a more general directed set. The directed set must be rich enough to "probe" all neighbourhoods of a point, which requires that any two elements have a common upper bound --- this is the property that $\mathbb{N}$ uses implicitly every time we write "for all $n \geq N$."

[definition: Directed Set] A **directed set** is a pair $(D, \preceq)$ where $D$ is a non-empty set and $\preceq$ is a binary relation on $D$ satisfying: 1. **Reflexivity:** $\alpha \preceq \alpha$ for all $\alpha \in D$. 2. **Transitivity:** If $\alpha \preceq \beta$ and $\beta \preceq \gamma$, then $\alpha \preceq \gamma$. 3. **Directedness:** For every $\alpha, \beta \in D$, there exists $\gamma \in D$ with $\alpha \preceq \gamma$ and $\beta \preceq \gamma$. A directed set is a preordered set (reflexive and transitive, but not necessarily antisymmetric) with the additional property that every pair of elements has an upper bound. [/definition]

The natural numbers $(\mathbb{N}, \leq)$ are the prototypical directed set, but the definition is far more permissive. The neighbourhood filter of a point, ordered by reverse inclusion, is a directed set --- and this is precisely the directed set that makes nets powerful in general topology.

[example: The Neighbourhood Directed Set] Let $(X, \tau)$ be a topological space and let $x \in X$. Define $\mathcal{N}(x) := \{U \in \tau : x \in U\}$, the collection of all open neighbourhoods of $x$. The relation \begin{align*} U \preceq V \quad :\Longleftrightarrow \quad V \subset U \end{align*} (reverse inclusion) makes $(\mathcal{N}(x), \preceq)$ into a directed set: reflexivity and transitivity are immediate from the corresponding properties of $\subset$, and directedness holds because $U \cap V$ is an open neighbourhood of $x$ with $U \preceq U \cap V$ and $V \preceq U \cap V$. In a metric space, the countable subfamily $\{B(x, 1/n) : n \in \mathbb{N}\}$ is cofinal in $\mathcal{N}(x)$ (every open neighbourhood of $x$ contains some $B(x, 1/n)$). This is why sequences, indexed by $\mathbb{N}$, suffice in metric spaces. In a non-first-countable space, $\mathcal{N}(x)$ has no countable cofinal subset, and $\mathbb{N}$ is too small to serve as an index set. [/example]

With directed sets in hand, the definition of a net is immediate.

[definition: Net] Let $(X, \tau)$ be a topological space and $(D, \preceq)$ a directed set. A **net** in $X$ is a function \begin{align*} s: D &\to X \\ \alpha &\mapsto s_\alpha. \end{align*} We write $(s_\alpha)_{\alpha \in D}$ for the net, in analogy with the sequence notation $(a_n)_{n \in \mathbb{N}}$. The net $(s_\alpha)_{\alpha \in D}$ **converges** to a point $x \in X$, written $s_\alpha \to x$, if for every open set $U$ containing $x$, there exists $\alpha_0 \in D$ such that $s_\alpha \in U$ for all $\alpha \succeq \alpha_0$. In this case, $x$ is called a **limit** of the net. [/definition]

When $D = \mathbb{N}$ with the usual ordering, a net is a sequence and this definition reduces to the standard definition of sequential convergence. In a [Hausdorff space](/page/Hausdorff%20Space), limits of nets are unique (the proof is identical to the sequential case: if $x \neq y$, choose disjoint open sets $U \ni x$ and $V \ni y$, then the net is eventually in both $U$ and $V$, contradicting directedness).

[remark: Nets vs Sequences --- What Changes] The passage from sequences to nets is not merely a matter of replacing $\mathbb{N}$ with a larger index set. There are several subtle differences: 1. **Uncountable index sets.** A net may be indexed by an uncountable directed set. The net "probing all neighbourhoods" of a point $x$ in a non-first-countable space requires an uncountable index set. 2. **Non-linear ordering.** A directed set need not be totally ordered. Two indices $\alpha$ and $\beta$ may be incomparable (neither $\alpha \preceq \beta$ nor $\beta \preceq \alpha$), so a net does not have a well-defined "order of evaluation." 3. **Non-antisymmetric ordering.** Different indices $\alpha \neq \beta$ may satisfy both $\alpha \preceq \beta$ and $\beta \preceq \alpha$, so the preorder on $D$ is not necessarily a partial order. Despite these differences, the "eventually" quantifier --- "there exists $\alpha_0$ such that for all $\alpha \succeq \alpha_0$..." --- works exactly as it does for sequences. [/remark]

## Characterising Topology via Nets

The central payoff of nets is that they recover the full sequential characterisations of closure, continuity, and compactness, without any first-countability hypothesis. In metric spaces, these characterisations are routine consequences of the countable neighbourhood base at each point. In general topological spaces, they require the richer indexing that nets provide.

### Closure

The [closure](/page/Closure) of a set $A$ in a metric space consists of all sequential limits from $A$. In a general topological space, the sequential closure may be strictly smaller than the topological closure, as we saw in the opening example. Nets repair this gap.

[quotetheorem:1045]