nLab preconvergence space

Idea

A preconvergence space is a generalisation of a convergence space or filter space which only the isotone axiom holds.

Definitions

A preconvergence space is a set $S$ together with a relation $\to$ from $\mathcal{F}S$ to $S$, where $\mathcal{F}S$ is the set of filters on $S$; if $F \to x$, we say that $F$ converges to $x$ or that $x$ is a limit of $F$. This must satisfy the axioms:

1. Isotone: If $F \subseteq G$ and $F \to x$, then $G \to x$;

The definition can also be phrased in terms of nets; a net $\nu$ converges to $x$ if and only if its eventuality filter converges to $x$.

The morphisms of preconvergence spaces are the continuous functions; a function $f$ between preconvergence spaces is continuous if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{f(A) \;|\; A \in F\}$. In this way, preconvergence spaces form a concrete category $PreConv$.

A preconvergence space that satisfies additional centred and directedness criterion is precisely a convergence space; see there for a variety of intermediate notions leading up to ordinary topological spaces.

See also

• limit of a filter
• Hausdorff space
• pointwise continuous function
• convergence space
• filter space

References

• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC) 945169917.

• Dolecki, Szymon (2009). Mynard, Frédéric; Pearl, Elliott (eds.). “An initiation into convergence theory”. Beyond Topology. Contemporary Mathematics Series A.M.S. 486: 115–162. (pdf)

• Dolecki, Szymon; Mynard, Frédéric (2014). “A unified theory of function spaces and hyperspaces: local properties”. Houston J. Math. 40 (1): 285–318. (pdf)

In the literature about uniform convergence, there is s the notion of preconvergence space as a specific kind of preuniform convergence space, which is different from the preconvergence spaces discussed above:

• Preuß, Gerhard. “Non-symmetric convenient topology and its relations to convenient topology.” Topology Proceedings, Volume 29, No.2, 2005, Pages 595-611, pdf.

• Fang, Jinming. “Lattice-valued preuniform convergence spaces.” Fuzzy Sets and Systems, vol. 251, Sept. 2014, pp. 52–70, doi:10.1016/j.fss.2013.11.010.

• Preuß, Gerhard. “Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions.” Topology and Its Applications, vol. 156, no. 12, July 2009, pp. 2005–2012, doi:10.1016/j.topol.2009.03.026.