nLab generalized filter space

Idea

A generalisation of various kinds of spaces equipped with a relation between its elements and its filters, such as filter spaces, convergence spaces, and cluster spaces.

WARNING: “Generalised filter space” is a placeholder name for a concept which may or may not have another name in the mathematics literature. If this concept already exists in the literature under another name, feel free to rename this page; alternatively, if “generalised filter space” is already being used for a different concept from the one described in this article, feel free to develop another name for this concept and move this article to that name.

Definition

Given a set $S$, let $\mathcal{F}(S)$ denote the set of filters on $S$. A set $S$ is a generalised filter space if it comes with a binary relation $x \to c$ between the $\mathcal{F}(S)$ and $S$ itself, for elements $x \in \mathcal{F}(S)$ and $c \in S$.

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 generalised filter space are the pointwise continuous functions; a function $f$ between generalised filter space is pointwise continuous if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A) \;|\; A \in F\}$.

According to the category theory Zulip, these structures are equivalently endofunctor algebras for the lax extension of the filter monad to Rel, the category of sets and relations.

See also

• limit of a filter
• Hausdorff space
• pointwise continuous function
• generalised sequential space
• centipede mathematics

 References

Some discussion about this mathematical structure happened on the category theory Zulip in:

• One Space to Rule Them All?, Category Theory Zulip. (web)