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 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 , a pregauge on is simply a family of pseudometrics on . A gauge is a -filter of pseudometrics on , that is a collection of gauging distances such that
for all in ; in the light of (3), the pseudometric is a gauging distance.
for all in , then .
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 and , a short map from to is a function (on their underlying sets) such that the composite with (or with , to be precise) of any gauging distance on is a gauging distance on . That is,
for all in ; in the light of (3), the pseudometric (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 of gauge spaces; quasigauge spaces and short maps between them form the category of quasigauge spaces. Note that any gauge is a base for a quasigauge; in this way, is (equivalent to) a full subcategory of .
The categories and 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 and are full subcategories of ; similarly, and are full subcategories of .
A uniform space defines a gauge space, consisting of all of the pseudometrics on that are uniformly continuous as maps from to the real line; similarly, a quasiuniform space defines a quasigauge space. Taking uniformly continuous maps as morphisms, the category is a full subcategory of ; similarly, is a full subcategory of . Thus, uniform spaces can be viewed as gauge spaces whose collection of gauging distances is “saturated” under uniform continuity.
A completely regular topological space defines a gauge space, consisting of all the pseudometrics on that are continuous as maps from to the real line. In this way the category of completely regular spaces and continuous maps is a full subcategory of ; it is in fact contained in . (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 in a space , define the pseudometric (in fact a pseudoultrametric) as follows:
then the form a base for a quasigauge, which induces the original topology on . In other words, every space is “quasigaugeable.” In this way also becomes a full subcategory of . Note, though that replacing by any other positive real number gives a different embedding of in .
Another way to get a quasigauge space from a topological space is to take as a base the set of all quasi-pseudometrics on such that for each , the function is upper semicontinuous. This is also a full embedding of in , which is perhaps more “canonical.”
Many of these full subcategories of and 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
Last revised on October 24, 2023 at 19:02:27. See the history of this page for a list of all contributions to it.