A gauge space is a topological space (necessary completely regular) whose topology is given by a family of pseudometrics. More generally, a quasigauge space is a space (not necessarily completely regular) whose topology is given by a family of quasipseudometrics.
Actually, a gauge space has additional structure, so that it can be seen as giving a (completely regular) Cauchy space, a uniform space, or even a generalisation of a metric space in which the category $Met$ of metric spaces and short maps is a full subcategory.
Please note that, while this is based on the presentation in HAF, the precise definitions of the objects and morphisms of the category of gauge spaces below constitute original research. (In particular, HAF really considers the category of pregauge spaces and uniformly continuous maps, which is equivalent to the category of uniform spaces, since it uses them only to study that category.)
Given a set $X$, a pregauge on $X$ is simply a family of pseudometrics on $X$. A gauge is a $\geq$-filter of pseudometrics on $X$, that is a collection $G$ of gauging distances such that
for all $x, y$ in $X$; in the light of (3), the pseudometric $( x, y \mapsto max(d(x,y), e(x,y)) )$ is a gauging distance.
for all $x, y$ in $X$, then $e \in G$.
A pregauge satisfying axioms (1&2) is a base for a gauge; a base is precisely what generates a gauge by taking the downward closure. Any pregauge whatsoever is a subbase for a gauge; a subbase is precisely what generates a base by closing under finitary joins.
A gauge space is a set equipped with a gauge. A quasigauge is a collection of quasipseudometrics satisfying (1–3); a quasigauge space is a set equipped with a quasigauge.
Given (quasi)gauge spaces $X$ and $X'$, a short map from $X$ to $X'$ is a function $F$ (on their underlying sets) such that the composite with $F$ (or with $F \times F$, to be precise) of any gauging distance on $X'$ is a gauging distance on $X$. That is,
for all $x, y$ in $X$; in the light of (3), the pseudometric $( x, y \mapsto e(F(x),F(y)) )$ (this is the composite) is a gauging distance.
(Warning: this definition of short map is probably the most significant original research on this page.)
Gauge spaces and short maps between them form the category $Gau$ of gauge spaces; quasigauge spaces and short maps between them form the category $QGau$ of quasigauge spaces. Note that any gauge is a base for a quasigauge; in this way, $Gau$ is (equivalent to) a full subcategory of $QGau$.
The categories $Gau$ and $QGau$ are not well known, but some of their subcategories are.
A metric space (or pseudometric space) defines a gauge space, taking its single (pseudo)metric as a base; similarly, a quasi(psuedo)metric space defines a quasigauge space. Taking short maps (in the usual sense, that is distance-nonincreasing functions) as morphisms, the categories $Met$ and $PsMet$ are full subcategories of $Gau$; similarly, $QMet$ and $QPsMet$ are full subcategories of $QGau$.
A uniform space $X$ defines a gauge space, consisting of all of the pseudometrics on $X$ that are uniformly continuous as maps from $X \times X$ to the real line; similarly, a quasiuniform space defines a quasigauge space. Taking uniformly continuous maps as morphisms, the category $Unif$ is a full subcategory of $Gau$; similarly, $QUnif$ is a full subcategory of $QGau$. Thus, uniform spaces can be viewed as gauge spaces whose collection of gauging distances is “saturated” under uniform continuity.
A completely regular topological space $X$ defines a gauge space, consisting of all the pseudometrics on $X$ that are continuous as maps from $X \times X$ to the real line. In this way the category $CReg Top$ of completely regular spaces and continuous maps is a full subcategory of $Gau$; it is in fact contained in $Unif$. (In general, a completely regular space can be uniformized in many ways; this inclusion corresponds to the “initial” uniformity.)
An arbitrary topological space defines a quasigauge space in a more complicated way. Given any open set $U$ in a space $X$, define the pseudometric (in fact a pseudoultrametric) $d_U$ as follows:
then the $d_U$ form a base for a quasigauge, which induces the original topology on $X$. In other words, every space is “quasigaugeable.” In this way $Top$ also becomes a full subcategory of $QGau$. Note, though that replacing $1$ by any other positive real number gives a different embedding of $Top$ in $QGau$.
Another way to get a quasigauge space from a topological space is to take as a base the set of all quasi-pseudometrics $d$ on $X$ such that for each $x\in X$, the function $d(x,-):X\to \mathbb{R}$ is upper semicontinuous. This is also a full embedding of $Top$ in $QGau$, which is perhaps more “canonical.”
Mike: Back atcha… do you have any good examples of gauge spaces that are not of one of these types? And is there any value in embedding metric spaces, uniform spaces, and topological spaces into this mysterious larger category $QGau$, in ways so that their images are essentially disjoint? Do we ever, for instance, want to talk about a short map from a metric space to a topological space, or vice versa? I would like the answer to be “yes,” but I haven’t managed to make it come out that way myself yet.
Toby: I don't know any examples of gauge spaces that arise naturally (you can make them artificially, of course, say as disjoint unions) and don't correspond to some other more familiar type of space. However, I can give examples that don't belong to $Met$, $Unif$, or $Top$ … because they're Cauchy spaces, which aren't listed above yet!
Incidentally, I didn't list Cauchy spaces yet (and didn't finish describing general topological spaces), since I haven't checked yet that things behave correctly. (In particular, I haven't checked that $Top$ becomes a full subcategory of $QGau$ —did you?—, although I hope that it will.)
I think that it's nice to be able to see these all as special cases of one kind of thing; then the usual ways of finding the ‘underlying’ foo of a bar become reflections (and sometimes coreflections). This is basically how Lowen-Colebunder's book (which I've referenced, for example, on convergence space) works (although her big category of everything is the category of mereological spaces, which includes $Conv$, rather than $QGau$).
Seeing this as inclusions and (co)reflections prevents any expectation that the diagrams above should commute, since they mix different things in the wrong way. On the other hand, it also shows us that (for example) any topological space has an underlying uniform space … if you want it.
Mike: I believe I did check that $Top\to QGau$ is full, but you should verify it.
I removed my comments about triangles commuting; go ahead and write about the reflections and I’ll see whether I still want to make that comment. I can see thinking of uniform spaces as “saturated” gauge spaces, so that the uniform space underlying a metric space is its “saturation” (reflection) in the larger category $Gau$. Perhaps Cauchy spaces can also be thought of this way. I will be kind of surprised if $Top$ turns out to be reflective in $QGau$, but if it were that would also be pretty neat.
Mike: I added a new embedding of $Top$ in $QGau$, which seems to me more likely to be (co)reflective.
Toby: Ah yes, I think that that's the good one that I wasn't finding.
Many of these full subcategories of $Gau$ and $QGau$ are reflective.
Details to come
Eric Schechter; 1997; Handbook of Analysis and its Foundations; Academic Press, 1st ed (1996).
Mike Shulman, The shape of infinity, arxiv