{"id":3554,"date":"2021-04-22T11:19:31","date_gmt":"2021-04-22T09:19:31","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=3554"},"modified":"2022-11-19T14:58:56","modified_gmt":"2022-11-19T13:58:56","slug":"bitopological-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=3554","title":{"rendered":"Bitopological spaces and stable compactness"},"content":{"rendered":"\n

A while back (in March 2019, to be precise), Tom\u00e1\u0161 Jakl<\/a> told me that he had a nice, short proof of the fact that the categories of stably compact spaces (and perfect maps) and compact pospaces (and continuous order-preserving maps) are equivalent [1]. He uses an approach through bitopological spaces, and this will give me an opportunity to talk about them.<\/p>\n\n\n\n

By the way, I am not claiming, and he is not claiming either, that he is the inventor of that proof. All the ingredients had been there for a long time. But it is worth to present them in a standalone fashion, as he did, with just as little of the theory of bitopological spaces as needed.<\/p>\n\n\n\n

So what is a bitopological space?<\/p>\n\n\n\n

Well, simply a set with two<\/em> topologies on it. Nothing more. Rather disappointing, at first sight. As simple as it may seem, the whole beauty of the notion is in the interaction between the two topologies. The main interaction we will study here is order separation<\/em>; this will be defined below.<\/p>\n\n\n\n

Bitopological spaces were introduced by J. C. Kelly in 1963 [2], and further studied by a number of authors. Let me mention Ralph Kopperman’s comprehensive paper on the subject [3].<\/p>\n\n\n\n

Examples of bitopological spaces<\/h2>\n\n\n\n

Let me write (X<\/em>, \u03c4+<\/sup>, \u03c4\u2013<\/sup><\/sub>) for a bitopological space: a set X<\/em>, and two topologies \u03c4+<\/sup> and \u03c4\u2013<\/sup><\/sub>. Trivially, (X<\/em>, \u03c4\u2013<\/sup><\/sub>, \u03c4+<\/sup>) is also a bitopological space, called the dual<\/em> of (X<\/em>, \u03c4+<\/sup>, \u03c4\u2013<\/sup><\/sub>). And the bidual, namely the dual of the dual, is the original bitopological space. Nothing to be amazed at, really, of course.<\/p>\n\n\n\n

Here are a few classical examples of (classes of) bitopological spaces.<\/p>\n\n\n\n

    \n
  1. (The bitopological space of a hemi-metric space) Let X<\/em>,d<\/em> be a hemi-metric space. There is an opposite hemi-metric d<\/em>op<\/sup> on X<\/em>, defined by d<\/em>op<\/sup>(x<\/em>,y<\/em>) \u225d d<\/em>(y<\/em>,x<\/em>). Then X<\/em> with the open ball topology of d<\/em> for \u03c4+<\/sup> and with the open ball topology of d<\/em>op<\/sup> for \u03c4\u2013<\/sup><\/sub> is a bitopological space.<\/li>\n\n\n\n
  2. (The bitopological space of a quasi-uniform space) Generalizing the previous example, let X<\/em> be a quasi-uniform space, with quasi-uniformity U<\/em><\/strong>. As we have seen in Part II<\/a> of the posts on quasi-uniform spaces, there is a dual quasi-uniformity U<\/strong><\/em>\u20131<\/sup>, whose entourages are the opposites R<\/em>\u20131<\/sup> (also written as R<\/em>op<\/sup>) of entourages R<\/em> in U<\/em><\/strong>. We may take the induced topology of U<\/em><\/strong> for \u03c4+<\/sup> and the induced topology of U<\/strong><\/em>\u20131<\/sup> for \u03c4\u2013<\/sup><\/sub>.<\/li>\n\n\n\n
  3. (The bitopological space of a pospace) Let (X<\/em>,\u2aaf) be a pospace. We obtain a bitopological space by letting \u03c4+<\/sup> be its upward topology (whose open sets are the open subsets of X<\/em> that are upwards closed with respect to \u2aaf) and \u03c4\u2013<\/sup><\/sub> be its downward topology (the open subsets that are downwards closed with respect to \u2aaf). We will be especially interested in the case where X<\/em> is compact.<\/li>\n\n\n\n
  4. (The Lawson bitopological space of a dcpo) Let X<\/em> be a dcpo. We take \u03c4+<\/sup> to be the Scott topology of X<\/em>, and \u03c4\u2013<\/sup><\/sub> to be its lower topology (whose open sets are unions of sets of the form \u2191E<\/em>, with E<\/em> finite). That is particularly interesting when X<\/em> is a continuous dcpo, or more generally a quasi-continuous dcpo.<\/li>\n<\/ol>\n\n\n\n

    Of course, those examples are far from random. In each case, the join \u03c4+<\/sup> \u2228 \u03c4\u2013<\/sup><\/sub> of the two topologies, namely the coarsest topology that contains both \u03c4+<\/sup> and \u03c4\u2013<\/sup><\/sub>, is an interesting topology in its own right.<\/p>\n\n\n\n