{"id":2869,"date":"2020-10-19T15:33:24","date_gmt":"2020-10-19T13:33:24","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2869"},"modified":"2022-11-19T15:01:12","modified_gmt":"2022-11-19T14:01:12","slug":"quasi-uniform-spaces-i-pervin-quasi-uniformities-pervin-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2869","title":{"rendered":"Quasi-Uniform Spaces I: Pervin quasi-uniformities, Pervin spaces"},"content":{"rendered":"\n

A uniform space<\/em> is a natural generalization of the notion of a metric space, on which completeness still makes sense [1]. The non-Hausdorff variant of this is called a quasi-uniform space<\/em>, and the purpose of this post is to introduce some of the features of those spaces, with a particular stress on a now very classical construction due to William Pervin [3]. It is rather puzzling that I managed to avoid the subject of quasi-uniform spaces in something like the 7 years that this blog existed!<\/p>\n\n\n\n

The definition of quasi-uniform spaces<\/h2>\n\n\n\n

Let X<\/em> be a set. A binary relation R<\/em> on X<\/em> is a subset of X<\/em> \u00d7 X<\/em>. Binary relations compose: R<\/em> o S<\/em> is the set of those pairs (x<\/em>,z<\/em>) such that (x<\/em>,y<\/em>) \u2208 R<\/em> and (y<\/em>,z<\/em>) \u2208 S<\/em> for some y<\/em> in X<\/em>. Note that R<\/em> o R<\/em> \u2286 R<\/em> if and only if R<\/em> is transitive. A relation R<\/em> is reflexive if and only if it contains the diagonal \u0394 \u225d {(x<\/em>,x<\/em>) | x<\/em> \u2208 X<\/em>}.<\/p>\n\n\n\n

A quasi-uniformity<\/em> U<\/em><\/strong> on X<\/em> is a filter of reflexive binary relations on X<\/em> that satisfies the following relaxed transitivity condition: for every relation S<\/em> in U<\/em><\/strong>, there is a relation R<\/em> in U<\/em><\/strong> such that R<\/em> o R<\/em> \u2286 S<\/em>. Explicitly, it is a family of binary relations on X<\/em> such that:<\/p>\n\n\n\n

    \n
  1. (reflexivity) for every R<\/em> in U<\/em><\/strong>, for every x<\/em> in X<\/em>, (x<\/em>,x<\/em>) is in R<\/em>;<\/li>\n\n\n\n
  2. (filter 1) X<\/em> \u00d7 X<\/em> is in U<\/em><\/strong>;<\/li>\n\n\n\n
  3. (filter 2) for every R<\/em> in U<\/em><\/strong>, every binary relation S<\/em> on X<\/em> such that R<\/em> \u2286 S<\/em> is in U<\/em><\/strong>;<\/li>\n\n\n\n
  4. (filter 3) for all R<\/em>, S<\/em> in U<\/em><\/strong>, R<\/em> \u2229 S<\/em> is in U<\/em><\/strong>;<\/li>\n\n\n\n
  5. (relaxed transitivity) for every S<\/em> in U<\/em><\/strong>, there is a relation R<\/em> in U<\/em><\/strong> such that, for all points x<\/em>, y<\/em>, z<\/em> in X<\/em> such that (x<\/em>,y<\/em>) \u2208 R<\/em> and (y<\/em>,z<\/em>) \u2208 R<\/em>, we have (x<\/em>,z<\/em>) \u2208 S<\/em>.<\/li>\n<\/ol>\n\n\n\n

    A quasi-uniform space<\/em> is a set with a quasi-uniformity U<\/em><\/strong>. The relations R<\/em> in U<\/em><\/strong> are called entourages<\/em>, a French word that means surroundings. (And also environment, neighborhood, relatives, acquaintances. As most French words, the word also exists in English.)<\/p>\n\n\n\n

    The primary example is given by hemi-metric spaces. Given any hemi-metric d<\/em> on X<\/em>, one defines a quasi-uniformity U<\/strong>d<\/sub><\/em> as follows. A basic entourage<\/em> is a relation of the form [<r<\/em>] \u225d {(x<\/em>,y<\/em>) | d<\/em>(x<\/em>,y<\/em>) < r<\/em>}, where r<\/em> varies over the positive real numbers (namely, r<\/em>>0). Then U<\/strong>d<\/sub><\/em> is the family of relations that contains some basic entourage [<r<\/em>].<\/p>\n\n\n\n

    It is interesting to see why relaxed transitivity holds. Let S<\/em> be in U<\/strong>d<\/sub><\/em>, so S<\/em> contains some basic entourage [<s<\/em>], with s<\/em>>0. Define r<\/em>\u225ds<\/em>\/2 and R<\/em> as [<r<\/em>]. For all points x<\/em>, y<\/em>, z<\/em> in X<\/em> such that (x<\/em>,y<\/em>) \u2208 R<\/em> and (y<\/em>,z<\/em>) \u2208 R<\/em>, we have d<\/em>(x<\/em>,y<\/em>)<r<\/em> and d<\/em>(y<\/em>,z<\/em>)<r<\/em>, so d<\/em>(x<\/em>,z<\/em>)\u2264d<\/em>(x<\/em>,y<\/em>)+d<\/em>(y<\/em>,z<\/em>)<s<\/em>, showing that (x<\/em>,z<\/em>) is in S<\/em>. In other words, relaxed transitivity stems from the triangular inequality.<\/p>\n\n\n\n

    By the way, the way we have built U<\/strong>d<\/sub><\/em> is a practical way of defining a quasi-uniformity. A base (of entourages<\/em>) is any non-empty family B<\/em><\/strong> of binary relations on X<\/em> satisfying:<\/p>\n\n\n\n

      \n
    1. (reflexivity) for every R<\/em> in B<\/em><\/strong>, for every x<\/em> in X<\/em>, (x<\/em>,x<\/em>) is in R<\/em>;<\/li>\n\n\n\n
    2. (filter 3) for all R<\/em>, S<\/em> in B<\/em><\/strong>, there is a relation T<\/em> in B<\/em><\/strong> such that T<\/em> \u2286 R<\/em> \u2229 S<\/em>;<\/li>\n\n\n\n
    3. (relaxed transitivity) for every S<\/em> in B<\/em><\/strong>, there is a relation R<\/em> in B<\/em><\/strong> such that R<\/em> o R<\/em> \u2286 S<\/em>.<\/li>\n<\/ol>\n\n\n\n

      Then the set of binary relations that contain some element of B<\/em><\/strong> forms a quasi-uniformity. This is the quasi-uniformity generated by<\/em> B<\/em><\/strong>. This is what we just did, with B<\/strong><\/em> consisting of the relations [<r<\/em>], r<\/em>>0.<\/p>\n\n\n\n

      Before we go on, I should mention the original notion of uniformity. Given any binary relation R<\/em> on X<\/em>, let R<\/em>op<\/sup> be its converse: (x<\/em>,y<\/em>) is in R<\/em>op<\/sup> if and only if (y<\/em>,x<\/em>) is in R<\/em>. A quasi-uniformity U<\/em><\/strong> is a uniformity<\/em> if and only if it satisfies the following extra condition:<\/p>\n\n\n\n

        \n
      1. (symmetry) for every R<\/em> in U<\/em><\/strong>, R<\/em>op<\/sup> is in U<\/em><\/strong>.<\/li>\n<\/ol>\n\n\n\n

        The prime example of a uniformity is U<\/strong>d<\/sub><\/em> provided that d<\/em> is a pseudo<\/em>-metric now.<\/p>\n\n\n\n

        Quasi-uniform spaces and topological spaces<\/h2>\n\n\n\n

        There is a traditional way to extract a topology from a quasi-uniformity U<\/strong><\/em>.<\/p>\n\n\n\n

        Given any point x<\/em> in X<\/em>, and any binary relation R<\/em> on X<\/em>, let R<\/em>[x<\/em>] denote the set {y<\/em> \u2208 X<\/em> | (x<\/em>,y<\/em>) \u2208 R<\/em>}; namely, the image<\/em> of x<\/em> by R<\/em>. We say that a U<\/strong>-neighborhood <\/em>of x<\/em> is a set of the form R<\/em>[x<\/em>] for some entourage R<\/em> \u2208 U<\/em><\/strong>. Next, we call U<\/strong>-open<\/em> any subset of X<\/em> that is a U<\/strong><\/em>-neighborhood <\/em>of each of its points. This forms a topology, which I will call the induced topology<\/em>, or the topology induced by<\/em> U<\/em><\/strong>. One also says that the quasi-uniformity U<\/em><\/strong> is compatible<\/em> with a topology if and only if that topology is the induced topology.<\/p>\n\n\n\n

        For example, if d<\/em> is a hemi-metric, a U<\/strong>d<\/sub><\/em>-neighborhood of x<\/em> is simply a subset of X<\/em> that contains the open ball Bd<\/sup>x<\/sub><\/em>,<r<\/em><\/sub> for some r<\/em>>0. The topology induced by U<\/strong>d<\/sub><\/em> is therefore the open ball topology of d<\/em>.<\/p>\n\n\n\n

        By the way, for every x<\/em> \u2208 X<\/em>, the U<\/strong>–<\/em>neighborhoods R<\/em>[x<\/em>] (R<\/em> \u2208 U<\/em><\/strong>) are neighborhoods of x<\/em> in the topology induced by U<\/em><\/strong>. Indeed, consider the set U<\/em> of all the points y<\/em> \u2208 X such that there is an S<\/em> \u2208 U<\/em><\/strong> such that S<\/em>[y<\/em>] is included in R<\/em>[x<\/em>]. This is open in the induced topology underlying U<\/em><\/strong>, by definition. In fact, it is easy to check that U<\/em> is the interior of R<\/em>[x<\/em>] in that topology. Additionally, x<\/em> is in U<\/em>, because one can take R<\/em> for S<\/em> in that case. Hence R<\/em>[x<\/em>] indeed contains an open set U<\/em> that contains x<\/em>.<\/p>\n\n\n\n

        A natural question arises: which topologies arise as topologies induced by quasi-uniformity? The question was solved by Cs\u00e1sz\u00e1r [2]: all topologies whatsoever! The simplest known argument is due to Pervin [3], and is as follows. There are in general several quasi-uniformities that induce the same topology, and we will see an example below; another way of stating the following theorem is to say that there is at least one.<\/p>\n\n\n\n

        Theorem (Cs\u00e1sz\u00e1r [2], Pervin [3]).<\/strong> Let X<\/em> be a topological space. For every open subset U<\/em> of X<\/em>, let RU<\/sub><\/em> be the binary relation {(x<\/em>,y<\/em>) | x<\/em> \u2208 U<\/em> \u21d2 y<\/em> \u2208 U<\/em>}. The finite intersections of relations RU<\/sub><\/em>, U<\/em> \u2208 O<\/strong>X<\/em>, form a base B<\/em><\/strong> of entourages of a quasi-uniformity on X<\/em>, whose induced topology is exactly the topology of X<\/em>. This quasi-uniformity is the Pervin quasi-uniformity<\/em> of the topology of X<\/em>.<\/p>\n\n\n\n

        Proof.<\/em> We check the axioms for a base of entourages. First, (reflexivity) and (filter 3) are clear. In order to verify (relaxed transitivity), we verify the much stronger statement that RU<\/sub><\/em> o RU<\/sub><\/em> \u2286 RU<\/sub><\/em> for every open subset U<\/em> of X<\/em>. That inclusion just means that if x<\/em> \u2208 U<\/em> \u21d2 y<\/em> \u2208 U<\/em> and if y<\/em> \u2208 U<\/em> \u21d2 z<\/em> \u2208 U<\/em>, then x<\/em> \u2208 U<\/em> \u21d2 z<\/em> \u2208 U<\/em>. The relation RU<\/sub><\/em> o RU<\/sub><\/em> \u2286 RU<\/sub><\/em> will play an important r\u00f4le later in this post.<\/p>\n\n\n\n

        We now consider an arbitrary element S<\/em> \u225d RU<\/sub><\/em>1<\/sub><\/sub> \u2229 … \u2229 RU<\/sub><\/em>n<\/em><\/sub><\/sub> in B<\/em><\/strong>. We look for an element R<\/em> of B<\/em><\/strong> such that R<\/em> o R<\/em> \u2286 S<\/em>, and we simply define R<\/em> as S<\/em>. Every pair (x<\/em>,z<\/em>) in R<\/em> o R<\/em> is such that there is a point y<\/em> such that (x<\/em>,y<\/em>) and (y<\/em>,z<\/em>) are in R<\/em>. Hence (x<\/em>,y<\/em>) and (y<\/em>,z<\/em>) are in RU<\/sub><\/em>i<\/em><\/sub><\/sub> for every i<\/em>, 1\u2264i<\/em>\u2264n<\/em>, which shows that (x<\/em>,z<\/em>) is in RU<\/sub><\/em>i<\/em><\/sub><\/sub> o RU<\/sub><\/em>i<\/em><\/sub><\/sub> \u2286 RU<\/sub><\/em>i<\/em><\/sub><\/sub> for every i<\/em>, 1\u2264i<\/em>\u2264n<\/em>, hence that (x<\/em>,z<\/em>) is in R<\/em>=S<\/em>.<\/p>\n\n\n\n

        Let U<\/em><\/strong> be the quasi-uniformity generated by B<\/em><\/strong>.<\/p>\n\n\n\n

        Given any point x<\/em> of X<\/em>, and any basic entourage S<\/em><\/em> \u225d RU<\/sub><\/em><\/em>1<\/sub><\/sub> \u2229 … \u2229 RU<\/sub><\/em>n<\/em><\/sub><\/sub><\/em>, the image S<\/em>[x<\/em>] is equal to the intersection of the opens sets Ui<\/sub><\/em><\/em> that contain x<\/em>. It follows that every U<\/em><\/strong>-open set is a neighborhood of each of its points, hence is open in the original topology on X<\/em>. Conversely, let U<\/em> be any open set in the original topology on X<\/em>. For every x<\/em> in U<\/em>, RU<\/sub><\/em><\/em>[x<\/em>]=U<\/em> is a U<\/em><\/strong>-neighborhood of x<\/em> included in U<\/em>, so U<\/em> is U<\/em><\/strong>-open. \u2610<\/p>\n\n\n\n

        You may have noticed the similarity with Wilson’s theorem (Theorem 6.3.13 in the book<\/a>), which says that every countably-based topological space is hemi-metrizable. You may have noticed that the proof strategy is very similar, too!<\/p>\n\n\n\n

        Indeed, each open subset U<\/em> of X<\/em> gives rise to a binary relation RU<\/sub><\/em>, but that relation has many properties. First and foremost, it is a preordering. Indeed, RU<\/sub><\/em> o RU<\/sub><\/em> \u2286 RU<\/sub><\/em>, as we have shown in the proof above, and that means that RU<\/sub><\/em> is transitive; and every entourage is reflexive by definition. Being a preordering, it induces a hemi-metric dU<\/sub><\/em>, defined by dU<\/sub><\/em>(x<\/em>,y<\/em>) \u225d 0 if (x<\/em>,y<\/em>) \u2208 RU<\/sub><\/em>, and 1 (or \u221e) otherwise. The open ball topology of that hemi-metric contains only U<\/em>, the empty set, and the whole space.<\/p>\n\n\n\n

        Now remember that the topology defined by a countable family of hemi-metrics is hemi-metrizable (Lemma 6.3.11). What we have just said shows the following.<\/p>\n\n\n\n

        Proposition.<\/strong> Every topology is defined by a family of hemi-metrics.<\/p>\n\n\n\n

        Proof. <\/em>Just take the hemi-metrics dU<\/sub><\/em>, where U<\/em> ranges over the open subsets. \u2610<\/p>\n\n\n\n

        As an aside, the topologies defined by a family of pseudo-metrics (symmetric hemi-metrics) are exactly the completely regular topologies, and are also exactly the topologies induces by uniformities. Let me allow not to say why this month, and to postpone it to a later post.<\/p>\n\n\n\n

        Morphisms<\/h2>\n\n\n\n

        At this point, one may be tempted to think that there is no point in inventing a new notion (quasi-uniform spaces), since quasi-uniform spaces and topological spaces seem to be one and the same thing.<\/p>\n\n\n\n

        The difference is in morphisms<\/em>. While the morphisms in the category Top<\/strong> of topological spaces are the continuous maps, the morphisms from X<\/em> to Y<\/em> in the category QUnif<\/strong> of quasi-uniform spaces are the uniformly continuous maps<\/em>, namely the maps f<\/em> : X<\/em> \u2192 Y<\/em> such that (f<\/em>\u00d7f<\/em>)\u20131<\/sup> (S<\/em>) is an entourage of X<\/em> for every entourage S<\/em> of Y<\/em>.<\/p>\n\n\n\n

        Every uniformly continuous map f<\/em> is continuous for the underlying induced topologies. Indeed, let V<\/em> be any open subset of Y<\/em>. For every x<\/em> \u2208 f<\/em>\u20131<\/sup>(V<\/em>), we need to show that there is a neighborhood of X<\/em> whose image by f<\/em> is included in V<\/em>. We use the fact that f<\/em>(x<\/em>) is in V<\/em>, and the definition of the induced topology on Y<\/em>, to obtain an entourage S<\/em> in Y<\/em> such that S<\/em>[f<\/em>(x<\/em>)] \u2286 V<\/em>. Let R<\/em> be the entourage (f<\/em>\u00d7f<\/em>)\u20131<\/sup> (S<\/em>). Then R<\/em>[x<\/em>] is a neighborhood of x<\/em>, by definition of the induced topology on X<\/em>, and every element of its image by f<\/em> is of the form f<\/em>(y<\/em>) with (x<\/em>,y<\/em>) \u2208 R<\/em>; so (f<\/em>(x<\/em>),f<\/em>(y<\/em>)) is in S<\/em>, by definition of R<\/em>, showing that f<\/em>(y<\/em>) is in S<\/em>[f<\/em>(x<\/em>)], hence in V<\/em>.<\/p>\n\n\n\n

        For quasi-metric spaces X<\/em> and Y<\/em>, with respective quasi-metrics dX<\/sub><\/em> and dY<\/sub><\/em>, a uniformly continuous map f<\/em> : X<\/em> \u2192 Y<\/em> between the corresponding quasi-uniform spaces is a map such that for every s<\/em>>0, (f<\/em>\u00d7f<\/em>)\u20131<\/sup> ([<s<\/em>]) contains a basic entourage [<r<\/em>] for some r<\/em>>0. In other words: for every s<\/em>>0, there is an r<\/em>>0 such that for all points x<\/em>, y<\/em> such that dX<\/sub><\/em>(x<\/em>,y<\/em>)<r<\/em>, dY<\/sub><\/em>(f<\/em>(x<\/em>),f<\/em>(y<\/em>))<s<\/em>. This is the usual definition of uniform continuity in (quasi-)metric spaces.<\/p>\n\n\n\n

        It is well-known that there are continuous maps between metric spaces that are not uniformly continuous, for example the map x<\/em> \u21a6 1\/x<\/em> on the positive real numbers. Hence that also holds in the realm of (quasi-)uniform spaces.<\/p>\n\n\n\n

        Rather remarkably, uniform continuity and continuity coincide on Pervin quasi-uniformities arising from topologies. Indeed, let X<\/em> and Y<\/em> be two topological spaces, and consider them with their Pervin quasi-uniformities. Let f<\/em> : X<\/em> \u2192 Y<\/em> be a continuous map. For every basic entourage S<\/em> \u225d RV<\/sub><\/em>1<\/sub><\/sub> \u2229 … \u2229 RV<\/sub><\/em>n<\/em><\/sub><\/sub> in Y<\/em>, (f<\/em>\u00d7f<\/em>)\u20131<\/sup> (S<\/em>) is the set of pairs (x<\/em>,y<\/em>) of points of X<\/em> such that f<\/em>(x<\/em>) \u2208 V<\/em>1<\/sub> implies f<\/em>(y<\/em>) \u2208 V<\/em>1<\/sub> and … and f<\/em>(x<\/em>) \u2208 V<\/em>n<\/em><\/sub> implies f<\/em>(y<\/em>) \u2208 V<\/em>n<\/em><\/sub>, in other words (f<\/em>\u00d7f<\/em>)\u20131<\/sup> (S<\/em>) is the basic entourage RU<\/sub><\/em>1<\/sub><\/sub> \u2229 … \u2229 RU<\/sub><\/em>n<\/em><\/sub><\/sub> where U<\/em>1<\/sub>\u225df<\/em>\u20131<\/sup>(V<\/em>1<\/sub>), …, U<\/em>n<\/em><\/sub>\u225df<\/em>\u20131<\/sup>(V<\/em>n<\/em><\/sub>).<\/p>\n\n\n\n

        What all that means, categorically, is:<\/p>\n\n\n\n