nLab cluster space

 Idea

The notion of clustering generalizes convergence.

Definition

A cluster space is a set $S$ together with a relation $\rightsquigarrow$ from $\mathcal{F}S$ to $S$; if $F \rightsquigarrow x$, we say that $F$ clusters at $x$ or that $x$ is a cluster point of $F$. The axioms are as follows:

1. Centred: The principal ultrafilter $F_x \rightsquigarrow x$;
2. Antitone: If $F \supseteq G$ and $F \rightsquigarrow x$, then $G \rightsquigarrow x$;
3. Codirected: If $F \cap G \rightsquigarrow x$ then $F \rightsquigarrow x$ or $G \rightsquigarrow x$.
4. Nontrivial: If $F \rightsquigarrow x$, then $F$ is proper.

Note that the direction of isotony and directedness for clustering is the reverse of that for convergence (hence ‘antitone’ and ‘codirected’). Nontriviality is the nullary version of directedness (equivalent to the statement that the improper filter never clusters at any point), which we explicitly need this time. Alternatively, we can take $\rightsquigarrow$ as a relation only on the proper filters; then nontriviality may be omitted from the axioms (as was done in the original reference, Muscat 2015).

Every convergence space is a cluster space (using the usual definition of $\rightsquigarrow$ from $\to$), and many of the notions of convergence generalize to cluster spaces, including continuous functions, open/closed sets, neighborhood filters, pre-closure, compactness, etc.

This definition of a cluster space does not seem to work in constructive mathematics. In particular, Muscat's proof that every convergence space satisfies the directedness axiom relies on excluded middle, and the lesser limited principle of omniscience follows if it holds in the real line, for example. It's not clear yet what if any alternative will work better.

Related concepts

• convergence space

References

• Joseph Muscat (2015). An axiomatization of filter clustering. Conference: 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD). DOI:10.1109/FSKD.2015.7381908.