{"id":1150,"date":"2017-02-23T18:51:07","date_gmt":"2017-02-23T17:51:07","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1150"},"modified":"2022-05-17T10:48:07","modified_gmt":"2022-05-17T08:48:07","slug":"topologies-on-spaces-of-lipschitz-yoneda-continuous-maps","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1150","title":{"rendered":"Topologies on Spaces of Lipschitz Yoneda-continuous Maps"},"content":{"rendered":"

Nota<\/strong> (added April 2, 2017): my interest in the following questions has faded.\u00a0 As I am saying at the end of this post, the questions I am asking here were a blocking point in trying to solve a problem on quasi-metrics for spaces of previsions.\u00a0 I have now found a way to solve the latter problem [3], and this does not require a complete solution to the questions of this page.\u00a0 Solving the questions on this page may offer a more elegant route, however.<\/p>\n

This is a tough question.\u00a0 I’ll have to do a bit of an introduction first.\u00a0 I will then give you a toned-down version of my question.\u00a0 If you manage to solve this one, maybe this will give us a lead into solving the real question.\u00a0 You may also want to look at the real question directly (in the third part).\u00a0 I will briefly say what the application of all that is in the final part of this post.<\/p>\n

Preliminaries<\/h2>\n

Let X<\/em>, d<\/em> be a metric space.\u00a0 Consider the set L<\/em>(X<\/em>) of all lower semi-continuous maps from X<\/em> to the extended non-negative reals [0, \u221e].\u00a0 This is also the set of all continuous maps from X<\/em> to [0, \u221e], where the latter comes with the Scott topology of \u2264.\u00a0 This can be given the Scott topology of the pointwise ordering, or the compact-open topology (understanding [0, \u221e] with its Scott topology).\u00a0 We have the following:<\/p>\n

Thm 1.<\/strong> [3, special case of Proposition 7.7] If X, d is a complete metric space, then the Scott topology and the compact-open topology coincide on L<\/em>(X<\/em>).<\/p>\n

We can now consider the subsets L\u03b1<\/sub><\/em>(X<\/em>) of those functions from X<\/em> to [0, \u221e] that are \u03b1-Lipschitz Yoneda-continuous, for every \u03b1>0.\u00a0 If you wonder about “Yoneda-continuous”, let me say that, because we are considering also the element \u221e in [0, \u221e], the topology on [0, \u221e], whether Scott or otherwise, is not the open ball topology of any metric or quasi-metric.\u00a0 Hence not all Lipschitz maps are continuous.\u00a0 I know that makes matters more complex.\u00a0 For a definition, see [1, Section 7.4.3]; you may wish to read Section 6 of [2], too.<\/p>\n

Finally, let me consider L\u221e<\/sub><\/em>(X<\/em>), the space of all Lipschitz Yoneda-continuous maps.\u00a0 This is the union of all the subspaces L\u03b1<\/sub><\/em>(X<\/em>).\u00a0 Give all those subspaces the subspace topology from L<\/em>(X<\/em>).\u00a0 As a consequence of Theorem 1, we have:<\/p>\n

Prop.<\/strong> Let X<\/em>, d<\/em> be a complete metric space.\u00a0 The spaces L\u221e<\/sub><\/em>(X<\/em>) and L\u03b1<\/sub><\/em>(X<\/em>) (\u03b1>0) all have the compact-open topology.<\/p>\n

That can be refined to:<\/p>\n

Thm 2.<\/strong>\u00a0 [3, special case of Proposition 8.1] Let X<\/em>, d<\/em> be a complete metric space.\u00a0 The spaces L\u03b1<\/sub><\/em>(X<\/em>) (\u03b1>0) all have the topology of pointwise convergence.<\/p>\n

The topology of pointwise convergence is the topology induced by the inclusion of L\u03b1<\/sub><\/em>(X<\/em>) into the product [0, \u221e]X<\/sup><\/em>, namely the space of all (not necessarily continuous) maps from X<\/em> to [0, \u221e].\u00a0 For a space of functions to [0, \u221e], the topology of pointwise convergence is always coarser than the compact-open topology, which is always coarser than the Scott topology.\u00a0 Here the first two coincide, and coincide with the subspace topology from L<\/em>(X<\/em>) with its own Scott topology.<\/p>\n

By using the fact that stable compactness under the formation of arbitrary products, under the extraction of closed subspaces, and under retracts, Theorem 2 also implies the following very nice result.<\/p>\n

Thm 3.<\/strong>\u00a0 [3, special case of Lemma 8.4 (4)] Let X<\/em>, d<\/em> be a complete metric space.\u00a0 Then the spaces L\u03b1<\/sub><\/em>(X<\/em>) (\u03b1>0) are stably compact.<\/p>\n

The toned-down version<\/h2>\n

Now here comes the delicate part.\u00a0 L\u03b1<\/sub><\/em>(X<\/em>) is a subspace of L\u221e<\/sub><\/em>(X<\/em>) for every \u03b1>0.\u00a0 I have certain functions F<\/em> : L\u221e<\/sub><\/em>(X<\/em>) \u2192 [0, \u221e] that I would like to show continuous (where [0, \u221e] still has the Scott topology), and I can show that their restrictions to L\u03b1<\/sub><\/em>(X<\/em>) are continuous for every \u03b1>0.<\/p>\n

That is not<\/em> enough to show that\u00a0F<\/em> itself is continuous!\u00a0 The problem is that the topology of L\u221e<\/sub><\/em>(X<\/em>) may fail to be determined by<\/em><\/a> the topologies of L\u03b1<\/sub><\/em>(X<\/em>).\u00a0 By definition, it is determined if and only if any subset U<\/em> of L\u221e<\/sub><\/em>(X<\/em>) whose intersection with L\u03b1<\/sub><\/em>(X<\/em>) is open in L\u03b1<\/sub><\/em>(X<\/em>) for each\u00a0\u03b1>0 is open in L\u221e<\/sub><\/em>(X<\/em>). In categorical terms: it is determined if and only if\u00a0L\u221e<\/sub><\/em>(X<\/em>) is a colimit of the diagram formed by the objects\u00a0L\u03b1<\/sub><\/em>(X<\/em>), with the obvious inclusions. Whence the question:<\/p>\n

Let X<\/em>, d<\/em> be a complete metric space.\u00a0 Is the topology of L\u221e<\/sub><\/em>(X<\/em>) determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0?<\/p><\/blockquote>\n

If it is not, then under which reasonable conditions is that topology determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0?\u00a0 Those conditions should be as general as possible.<\/p>\n

I already know that the topology is indeed determined if X<\/em>, d<\/em> is what I am currently calling Lipschitz regular<\/em> [3, Section 4].\u00a0 I will give the actual definition later.\u00a0 In the current context, it is perhaps simpler to say that a complete metric space is Lipschitz regular if and only if, writing B(x<\/em>, <r<\/em>) for the open ball of center x<\/em> and radius r<\/em>, the following property holds: for all positive reals\u00a0r<\/em> and\u00a0s<\/em> such that r<s<\/em>, B(x<\/em>, <r<\/em>) is relatively compact in B(x<\/em>, <s<\/em>) (i.e., way-below in the lattice of open sets; i.e., every open cover of B(x<\/em>, <s<\/em>) contains a finite subcover of B(x<\/em>, <r<\/em>)).<\/p>\n

In case X<\/em>, d<\/em> is Lipschitz regular, L\u03b1<\/sub><\/em>(X<\/em>) is in fact a retract, by an embedding-projection pair, of L\u221e<\/sub><\/em>(X<\/em>), where the embedding is just inclusion.\u00a0 That implies that the topology of L\u221e<\/sub><\/em>(X<\/em>) is determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0 [3, Proposition 9.2], but is a stronger property.<\/p>\n

Lipschitz regularity implies local compactness, and that excludes Baire space, for example.\u00a0 Hence I am not entirely pleased with that assumption.\u00a0 You may also want to show that any complete metric space X<\/em>, d<\/em> such that the topology of L\u221e<\/sub><\/em>(X<\/em>) is determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0, is in fact Lipschitz regular.\u00a0 That would give me an excuse for the notion.<\/p>\n

The real question<\/h2>\n

A quasi-metric space is a space X<\/em> with a quasi-metric<\/em> d<\/em>, namely a kind of metric except that d<\/em>(x<\/em>,y<\/em>) is not required to be equal to d<\/em>(y<\/em>,x<\/em>).\u00a0 There is a lot of information about those spaces in [1], chapters 6 and 7.\u00a0 You should also probably read [2] if you are interested.<\/p>\n

Theorem 1 is not what I really proved.\u00a0 What I proved is the following more general result:<\/p>\n

Thm 1′.<\/strong> [3, Proposition 7.7] If X, d is a continuous Yoneda-complete quasi-metric space, then the Scott topology and the compact-open topology coincide on L<\/em>(X<\/em>).<\/p>\n

This includes Theorem 1 as a special case, since complete metric spaces are Yoneda-complete, and all metric spaces are continuous.<\/p>\n

Thm 2′ and 3′.<\/strong>\u00a0 [3, Proposition 8.1, Lemma 8.4] Let X<\/em>, d<\/em> be a complete metric space.\u00a0 Then the spaces L\u03b1<\/sub><\/em>(X<\/em>) (\u03b1>0) all have the topology of pointwise convergence, and are stably compact.<\/p>\n

Accordingly, the real question is:<\/p>\n

Let X<\/em>, d<\/em> be a continuous Yoneda-complete metric space.\u00a0 Is the topology of L\u221e<\/sub><\/em>(X<\/em>) determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0?<\/p><\/blockquote>\n

If it is not, then under which reasonable conditions is that topology determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0?\u00a0 Those conditions should be as general as possible.\u00a0 Strengthening the assumption from “continuous Yoneda-complete” to “algebraic Yoneda-complete” is perfectly reasonable, and Section 7 of [2] gives a indication how to reduce the continuous case to the algebraic case.<\/p>\n

The actual definition of Lipschitz regular is the following.\u00a0 Let me consider that X<\/em> embeds into its space of formal balls B<\/strong>(X<\/em>,d<\/em>), through the map x<\/em> \u21a6 (x<\/em>, 0).\u00a0 In other words, let me equate x<\/em> with (x<\/em>, 0). \u00a0There is function that maps every d<\/em>-Scott open subset of U<\/em> to the largest Scott-open subset V<\/em> of B<\/strong>(X<\/em>,d<\/em>) such that V<\/em> \u2229 X<\/em> = U<\/em>.\u00a0 The space X<\/em>, d<\/em> is Lipschitz regular<\/em> if and only if that map is Scott-continuous.<\/p>\n

I already know that if X<\/em>, d<\/em> is Lipschitz regular in that sense, then the topology of L\u221e<\/sub><\/em>(X<\/em>) determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0; in fact L\u221e<\/sub><\/em>(X<\/em>) is obtained as a limit of a diagram of embedding-projection pairs, which is a stronger property.<\/p>\n

The purpose of all that<\/h2>\n

There is a famous theorem in measure theory that says that the space of all probability measures on a Polish space is again Polish, due to Prohorov.\u00a0 Concretely, that involves assuming a separable complete metric space X<\/em>, d<\/em>, and exhibiting a metric on the space of probabilities on X<\/em> that makes it separable and complete.<\/p>\n

Such metrics include the L\u00e9vy-Prohorov metric and the Kantorovitch-Rubinshtein (also called Hutchinson) metric.\u00a0 Completeness is a pretty tough theorem, and is usually proved by arguments on notions of tightness of measures, and of sets of measures.<\/p>\n

What I am trying to do is to generalize that to certain classes of quasi<\/em>-metric spaces.\u00a0 It turns out that the space of continuous valuations on X<\/em> (roughly the same as measures) is isomorphic to a certain space of continuous functionals F : L\u221e<\/sub>(X)<\/em> \u2192 [0, \u221e].\u00a0 One gets the functional from the measure by integrating the function given as argument against the measure.<\/p>\n

In an attempt to show that that space of functionals is (Yoneda-)complete, I used to be blocked by the fact that I can show that a certain functional F<\/em> is continuous from L\u03b1<\/sub><\/em>(X<\/em>) to [0, \u221e] for every \u03b1>0, but that I need it to be continuous on the whole of L\u221e<\/sub><\/em>(X<\/em>).\u00a0 I have managed to get around this problem by showing that the space of formal balls of X<\/em>, d<\/em>, is always Lipschitz regular, and that every continuous Yoneda-complete space embeds as a G<\/em>\u03b4 subset of its space of formal balls, then using measure extension theorems on continuous dcpos… what a mess.\u00a0 As I said, if the topology of L\u221e<\/sub><\/em>(X<\/em>) is determined by the topologies of L\u03b1<\/sub><\/em>(X<\/em>), \u03b1>0, there would be a much simpler proof available.<\/p>\n

Any ideas?<\/p>\n

    \n
  1. Jean Goubault-Larrecq. Non-Hausdorff Topology and Domain Theory \u2014 Selected Topics in Point-Set Topology<\/a>. New Mathematical Monographs 22. Cambridge University Press, 2013.<\/li>\n
  2. Jean Goubault-Larrecq and Kok Min Ng. A few notes on formal balls<\/a>. Logical Methods in Computer Science 13(4), nov. 28, 2017.<\/li>\n
  3. Jean Goubault-Larrecq.\u00a0 Complete quasi-metrics for hyperspaces, continuous valuations, and previsions<\/a>. arXiv 1707.03784, version 3, 28 oct. 2017.<\/li>\n<\/ol>\n

    \u2014 Jean Goubault-Larrecq<\/a>\u00a0(February 23rd, 2017)\"jgl-2011\"<\/p>\n","protected":false},"excerpt":{"rendered":"

    Nota (added April 2, 2017): my interest in the following questions has faded.\u00a0 As I am saying at the end of this post, the questions I am asking here were a blocking point in trying to solve a problem on … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-1150","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1150","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1150"}],"version-history":[{"count":5,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1150\/revisions"}],"predecessor-version":[{"id":5422,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1150\/revisions\/5422"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}