First, season’s greetings to you all!<\/p>\n\n\n\n
One of the big thing about uniform spaces is that they have a notion of completeness, and a notion of completion, which generalize those which are available in metric spaces. A similar thing can be said for quasi-uniform spaces<\/a>, too. However, remember that the situation was already complicated in quasi-metric spaces, where we have two notions of completeness, Smyth-completeness and Yoneda-completeness, one notion of completion (Yoneda-completion or the formal ball completion, giving isomorphic spaces), that the completion of a quasi-metric space is always Yoneda-complete and even algebraic Yoneda-complete, but that a quasi-metric space is isomorphic to its completion if and only if it is Smyth-complete. All those facts about quasi-metric spaces are in Chapter 7 of the book<\/a>.<\/p>\n\n\n\n In today’s post, we will explore natural quasi-uniform generalizations of the notions of Smyth-completeness, symcompactness, and total boundedness. My plan is to show that the following two theorems from the book<\/a> generalize to quasi-uniform spaces: every Smyth-complete quasi-metric space is sober, and the symcompact quasi-metric spaces are exactly the quasi-metric spaces that are Smyth-complete and totally bounded. The second one will be rather more intricate to prove than the first one, and there are new complications that do not arise in the quasi-metric case: be warned!<\/p>\n\n\n\n A quasi-metric space X<\/em>, d<\/em> is Smyth-complete if and only if every Cauchy net (xi<\/sub><\/em>)i<\/em> \u2208 I<\/em>, \u2291<\/sub> has a limit in the symmetrized space X<\/em>, d<\/em>sym<\/sup>. And a net (xi<\/sub><\/em>)i<\/em> \u2208 I<\/em>, \u2291<\/sub> is Cauchy if and only if for every for every r<\/em>>0, there is an i<\/em>0<\/sub> \u2208 I<\/em> such that for all i<\/em>, j<\/em> in I<\/em> such that i<\/em>0<\/sub> \u2291 i<\/em> \u2291 j<\/em>, d<\/em>(xi<\/sub><\/em>,xj<\/sub><\/em>) < r<\/em>.<\/p>\n\n\n\n In other words, a net (xi<\/sub><\/em>)i<\/em> \u2208 I<\/em>, \u2291<\/sub> is Cauchy if and only if for every for every r<\/em>>0, there is an i<\/em>0<\/sub> \u2208 I<\/em> such that for all i<\/em>, j<\/em> in I<\/em> such that i<\/em>0<\/sub> \u2291 i<\/em> \u2291 j<\/em>, (xi<\/sub><\/em>,xj<\/sub><\/em>) is in the basic entourage [<r<\/em>] of the quasi-uniformity defined by d<\/em>.<\/p>\n\n\n\n Generalizing, this, we can therefore define Cauchy nets in arbitrary quasi-uniform spaces by: a net (xi<\/sub><\/em>)i<\/em> \u2208 I<\/em>, \u2291<\/sub> in a quasi-uniform space X<\/em>, with quasi-uniformity U<\/em><\/strong>, is Cauchy<\/em> if and only if for every entourage R<\/em> in U<\/em><\/strong>, there is an i<\/em>0<\/sub> \u2208 I<\/em> such that for all i<\/em>, j<\/em> in I<\/em> such that i<\/em>0<\/sub> \u2291 i<\/em> \u2291 j<\/em>, (xi<\/sub><\/em>,xj<\/sub><\/em>) is in R<\/em>. Then, on quasi-metric spaces, a net is Cauchy in the quasi-metric sense if and only if it is Cauchy in the new, quasi-uniform sense.<\/p>\n\n\n\n Generalizing the notion of symmetrized space is even easier. Given a quasi-uniformity U<\/em><\/strong> on X<\/em>, 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 have seen that last time<\/a>. There is also a symmetrized<\/em> quasi-uniformity U<\/strong><\/em>sym<\/sup>, obtained as the smallest quasi-uniformity that contains both U<\/em><\/strong> and U<\/strong><\/em>\u20131<\/sup>. Explicitly, a base of entourages of U<\/strong><\/em>sym<\/sup> is given by the intersections R<\/em> \u2229 S<\/em>\u20131<\/sup> of an entourage R<\/em> of U<\/em><\/strong> and the opposite of an entourage S<\/em> of U<\/em><\/strong>. It is easy to see that another, smaller base of entourages is given by the symmetrizations R<\/em> \u2229 R<\/em>\u20131<\/sup> of entourages R<\/em> of U<\/em><\/strong>. In both cases, the conditions<\/a> (reflexivity), (filter3) and (relaxed transitivity) of bases of entourages are easy to check; the two bases contain the whole of U<\/em><\/strong> and of U<\/em><\/strong>\u20131<\/sup>; and any quasi-uniformity containing both U<\/em><\/strong> and U<\/em><\/strong>\u20131<\/sup> must contain each of those bases.<\/p>\n\n\n\n The quasi-uniformity U<\/strong><\/em>sym<\/sup> is symmetric, in the sense that the opposite of any entourage of U<\/strong><\/em>sym<\/sup> is again in U<\/strong><\/em>sym<\/sup>. Hence U<\/strong><\/em>sym<\/sup> is a uniformity. When U<\/em><\/strong> is the quasi-uniformity of a quasi-metric d<\/em>, a base of entourages of U<\/strong><\/em>sym<\/sup> is given by entourages of the form [<r<\/em>] \u2229 [<r<\/em>]\u20131<\/sup>, r<\/em>>0, which are just the basic entourages of the uniformity defined by the symmetrized metric d<\/em>sym<\/sup>.<\/p>\n\n\n\n Hence the notion of Smyth-completeness extends to quasi-uniform spaces as follows: a quasi-uniform space X<\/em>, with quasi-uniformity U<\/em><\/strong>, is Smyth-complete<\/em> if and only if every Cauchy net in X<\/em> has a limit in X<\/em>sym<\/sup>, where X<\/em>sym<\/sup> is X<\/em> with the (topology induced by the) uniformity U<\/strong><\/em>sym<\/sup>.<\/p>\n\n\n\n Let us put the notion to the test.<\/p>\n\n\n\nSmyth-completeness<\/h2>\n\n\n\n
Smyth-complete implies quasi-sober<\/h2>\n\n\n\n