A pretopological space is a slight generalisation of a topological space where the concept of neighbourhood is taken as primary. The extra structure on the underlying set of a pretopological space is called its pretopology, but this should not be confused with a Grothendieck pretopology (which is not even analogous).
Given a set , let a point in be an element of , and let a set in be a subset of . Given a relation between points in and sets in , say that the set is a neighbourhood (or -neighbourhood to be precise) of if (which may also be written ).
A pretopology (or pretopological structure) on is such a relation that satisfies these properties:
(Some references leave this out, but that seems to be an error.)
(Strictly speaking, the relation should not be called directed unless it is also nontrivial.)
A pretopology can also be given by a base or subbase. A base for a pretopology is any relation that satisfies (1–3), using the first version for each of (2,3); a subbase is any relation that satisfies (1). (It would not really be appropriate to use the symbol ‘’ for a mere base or subbase; you'd probably want to think of it as a family of sets indexed by the points, and use the term ‘basic neighourhood’ or ‘subbasic neighbourhood’.) You get a base (in fact one satisfying the stronger versions of 2,3) from a subbase by closing under finitary intersections; you get a pretopology from a base by taking supsersets. (Really, this is just a special case of considering a base or subbase of a filter.)
A pretopological space is a set equipped with a pretopological structure.
A continuous map from a pretopological space to a pretopological space is a function from to such that:
In this way, pretopological spaces and continuous maps form a category .
If is a filter on a pretopological space , then converges to a point (written ) if refines (contains) the neighbourhood filter of .
This relation satisfies the following properties:
In this way, every pretopological space becomes a convergence space.
In fact, we can recover the pretopological structure from the convergence structure as follows: if and only if belongs to every filter that converges to . In other words, that intersection that appears in the infinite filtration condition is the neighbourhood filter of . Furthermore, this definition assigns a pretopological structure to any convergence space satsifying the conditions above, and a map between pretopological spaces is continuous if and only if it is continuous as a map between convergence spaces. Thus, we can define a pretopological space as an infinitely directed convergence space, making a full subcategory of the category of convergence spaces.
Actually, we can do more. The definition of from the convergence structure assigns a pretopological structure to any convergence space, although in general this pretopology defines a weaker notion of convergence (more filters converge to more points). Thus, is also a reflective subcategory of .
Every pretopological convergence satisfies the star property, so that is a full reflective subcategory of the category of pseudotopological spaces.
Every topological space is a pretopological space, using the usual definition of (not necessarily open) neighbourhood: if there exists some open set such that and . Also, a map between topological spaces is continuous if and only if it's continuous as a map between pretopological spaces. In this way, the category Top of topological spaces becomes a full subcategory of .
In fact, we can easily characterise the topological pretopologies, allowing us to define a topological space as a pretopological space satisfying this axiom:
In the terms defined below, a topological space is a pretopological space in which every preinterior is open.
Here is an example of a nontopological pretopological space, although admittedly it is a bit artificial. (This is based on Section 15.6 of HAF.) Consider a metric space ; according to the usual pretopology on , is a neighbourhood of if there is a positive number such that contains the ball . Now given a natural number , we will give the plus pretopology: is a neighbourhood of if there is a positive number such that contains the -ball . (If is a line and , then this neighbourhood is a plus sign ‘+’ with at the centre and cross bars of length .) Then is a pretopological space, but it is topological only if or is a subsingleton.
This example can probably be generalised to a uniform space . Possibly there is some interesting universal property of this ‘plus product’, although it seems to go from to , so maybe we need to work in a different category. (There is a notion of uniform convergence space that generalises uniform spaces much like convergence spaces generalise topological spaces; perhaps the plus product takes place there.)
Fix a pretopological space .
The preinterior of a set is the set or of all points that is a neighbourhood of:
A set is open if it equals its preinterior. The interior of is the union of all of the open sets contained in . Note that we can immediately recover the pretopological structure from the preinterior operation (but not from the interior operation nor from the class of all open sets).
Similarly, the preclosure of is the set of all points that meets every neighbourhood of:
A set is closed if it equals its preclosure. The closure of is the intersection of all of the closed sets containing . Again, we can recover the pretopological structure from the preclosure operation; iff meets every set such that . (This result seems to require excluded middle.)
(Warning: not all references use these terms in the same way. This terminology is based on the premise that a closure should be closed.)
The duality between (pre)interiors and open sets on the one hand and (pre)closures and closed sets on the other hand is (at least if you assume excluded middle) just what you would expect: the (pre)interior of a complement is the complement of the (pre)closure, and a set is open if and only if its complement is closed. However, a preinterior is generally not open but larger than an interior; similarly, a preclosure is generally not closed but smaller than a closure. The situation looks like this:
In many cases this iteration stabilizes after finitely many terms. The plus power seems to stabilise after iterations. And in a topological space, of course, it only takes one step.
In general, however, there can be transfinitely many terms in these sequences. For example, let be any ordinal number (thought of as the well-ordered set of all smaller ordinal numbers) with the following pretopology:
Let . Then , , and so on, the process taking steps to stabilize at .
Note that an interior is open, and a closure is closed. Indeed, the open sets in form a topological structure on , giving the usual meanings of interior, closure, and closed set. This topological structure does not (in general) give the original pretopology on ; instead, this makes a reflective subcategory of .
In the definition of pretopology, the neighbourhoods of each point may be given completely independently of any other point. So the notion of topological space may also be seen as requiring some coherence between the neighbourhoods of nearby points.