Mathematics.

Convergence in General Topology

Nets and Filters

Topology75 minDifficulty8 out of 10

Overview

Sequences are sufficient to describe convergence in metric spaces, but in general topological spaces they fall short: a point may be in the closure of a set without any sequence in the set converging to it. Nets (Moore-Smith sequences) and filters generalise sequences to arbitrary topological spaces, restoring all the familiar sequential intuitions: a set is closed iff it is closed under limits of converging nets, a function is continuous iff it preserves net convergence, and a space is compact iff every net has a convergent subnet. Filters provide an equivalent, more algebraic formulation.

Intuition

In a metric space, sequences are fine because the neighbourhood system at each point has a countable basis. In a general topological space, neighbourhoods can be uncountable, and sequences miss too many of them. A net is like a sequence but indexed by any directed set — think of a 'generalised sequence' that can probe every neighbourhood simultaneously, no matter how large. Filters are a dual language: instead of tracking the index, a filter tracks which sets contain the 'eventual values' of the net.

Formal Definition

Definition

A directed set is a nonempty set \Lambda with a preorder \leq that is upward-directed: \forall \alpha, \beta \in \Lambda, \exists \gamma \geq \alpha, \beta.

A net in X is a function (xα)α∈Λ:Λ→X indexed by a directed set Λ.\text{A net in } X \text{ is a function } (x_\alpha)_{\alpha \in \Lambda}: \Lambda \to X \text{ indexed by a directed set } \Lambda.
Net
xα→x  ⟺  ∀U∋x open, ∃α0, α≥α0  ⟹  xα∈U.x_\alpha \to x \iff \forall U \ni x \text{ open},\ \exists \alpha_0,\ \alpha \geq \alpha_0 \implies x_\alpha \in U.
Net convergence
F is a filter on X  ⟺  (i)∅∉F, X∈F; (ii)A,B∈F  ⟹  A∩B∈F; (iii)A∈F,A⊆B  ⟹  B∈F.\mathcal{F} \text{ is a filter on } X \iff (\text{i}) \emptyset \notin \mathcal{F},\ X \in \mathcal{F};\ (\text{ii}) A,B \in \mathcal{F} \implies A \cap B \in \mathcal{F};\ (\text{iii}) A \in \mathcal{F}, A \subseteq B \implies B \in \mathcal{F}.
Filter axioms
F→x  ⟺  ∀U∋x open, U∈F.\mathcal{F} \to x \iff \forall U \ni x \text{ open},\ U \in \mathcal{F}.
Filter convergence

Properties

Characterisation of closure

x∈A‾  ⟺  ∃ a net in A converging to x.x \in \overline{A} \iff \exists \text{ a net in } A \text{ converging to } x.

Characterisation of continuity

f:X→Y is continuous  ⟺  for every net xα→x in X, f(xα)→f(x) in Y.f: X \to Y \text{ is continuous} \iff \text{for every net } x_\alpha \to x \text{ in } X,\ f(x_\alpha) \to f(x) \text{ in } Y.

Compactness via nets

X is compact  ⟺  every net in X has a convergent subnet.X \text{ is compact} \iff \text{every net in } X \text{ has a convergent subnet.}

Ultrafilters and compactness

X is compact  ⟺  every ultrafilter on X converges.X \text{ is compact} \iff \text{every ultrafilter on } X \text{ converges.}

Worked Examples

  1. (\Rightarrow) If F is closed, then X \setminus F is open. If x \notin F, then x \in X \setminus F and since x_\alpha \to x, eventually x_\alpha \in X \setminus F, contradicting the net being in F. So x \in F.

  2. (\Leftarrow) Suppose F is net-closed. We show X \setminus F is open. Let x \in X \setminus F. If every open neighbourhood U of x meets F, then the net (x_U)_{U \ni x} (ordered by reverse inclusion, with x_U \in U \cap F) converges to x. By net-closedness x \in F — contradiction. So some open U \subseteq X \setminus F, showing X \setminus F is open.

    Directed by: U≤V  ⟺  V⊆U.\text{Directed by: } U \leq V \iff V \subseteq U.

Answer: A set is closed iff it is closed under net limits — this is the net version of the sequential criterion for closed sets, valid in all topological spaces.

Practice Problems

Difficulty 6/10

What is a directed set? Give two examples, one that is and one that is not totally ordered.

Difficulty 7/10

Show that f: X \to Y is continuous iff for every net x_\alpha \to x in X, f(x_\alpha) \to f(x) in Y.

Difficulty 8/10

Define an ultrafilter and state its connection to compactness.

Common Mistakes

Common Mistake

Sequences suffice to characterise topology in any space.

Sequences suffice only in first-countable (e.g. metrizable) spaces. In general, nets or filters are needed.

Common Mistake

Every filter is an ultrafilter.

Ultrafilters are maximal filters. The neighbourhood filter at a point is not an ultrafilter in general; ultrafilters are obtained by extending via Zorn's lemma.

Quiz

Why are nets needed instead of sequences in general topology?
A filter on X is a collection of subsets that is:

Summary

  • Nets generalise sequences to arbitrary directed index sets, restoring sequential intuitions in all topological spaces.
  • Filters provide a dual, set-based language for convergence: a filter converges to x if every neighbourhood of x is in the filter.
  • Closure, continuity, and compactness all admit clean characterisations via nets/filters in any topological space.
  • Ultrafilters are maximal filters; a space is compact iff every ultrafilter converges.
  • Nets and sequences agree in first-countable (e.g. metrizable) spaces.

References

  1. BookKelley, J.L. — General Topology, Springer, 1975, Chapter 2.
  2. WebsiteWikipedia — Net (mathematics)