{"id":1503,"date":"2018-06-21T18:55:54","date_gmt":"2018-06-21T16:55:54","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1503"},"modified":"2022-11-19T15:16:08","modified_gmt":"2022-11-19T14:16:08","slug":"dcpos-and-convergence-spaces-ii-preserving-products","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1503","title":{"rendered":"Dcpos and convergence spaces II: preserving products"},"content":{"rendered":"

Let us continue last month’s story. \u00a0We had define various structures of convergence spaces on a dcpo, which were all admissible in the sense that their topological modification is the Scott topology.<\/p>\n

Amongst those,\u00a0Heckmann convergence<\/em> is defined by\u00a0limH\u00a0<\/sub>F<\/em> = cl (\u222a {A<\/em>\u2193<\/sup> | A<\/em> \u2208\u00a0F<\/em>}), for every filter\u00a0F<\/em> of subsets of\u00a0a dcpo\u00a0X<\/em>, where cl is Scott closure. \u00a0We had seen that\u00a0Scott convergence<\/em> was instead defined by\u00a0limS<\/sub> F\u00a0<\/em>= adh (\u222a {A<\/em>\u2193<\/sup> | A<\/em> \u2208\u00a0F<\/em>}), where\u00a0adh(B<\/em>) is the set of elements below suprema of directed families included in the downwards-closed subset\u00a0B<\/em>\u2014in general closure is obtained by iterating adh transfinitely.<\/p>\n

The case of products<\/h2>\n

There is a very nasty pitfall concerning products in domain theory: in general, the product of two dcpos as posets is different from their product as topological spaces. \u00a0Explicitly, let us write Z<\/em>\u03c3<\/sub> for a poset\u00a0Z<\/em> seen as a topological space with the Scott topology. \u00a0The pitfall is that if you take two dcpos\u00a0X<\/em> and\u00a0Y<\/em>, build their product X<\/em> x\u00a0Y<\/em>\u00a0as dcpos, and then take the Scott topology (X<\/em> x\u00a0Y<\/em>)\u03c3,\u00a0<\/sub>then what you will get will of course have the same points as the topological product X<\/i>\u03c3<\/sub>\u00a0x Y<\/em>\u03c3<\/sub>, but the topology will in general be different (see Exercise 5.2.16 in the book<\/a>).<\/p>\n

This has apparently caught a few people, including myself\u2014and despite me knowing of the problem.<\/p>\n

That problem does not occur with continuous dcpos, or even continuous posets.<\/p>\n

One nice observation due to\u00a0Reinhold Heckmann<\/a>\u00a0[1] is that the problem disappears completely once you see dcpos as convergence spaces, using Heckmann convergence, instead of topological spaces.<\/p>\n

Let us investigate that. \u00a0Let us write\u00a0Z<\/em>H<\/sub> for a dcpo\u00a0Z<\/em> equipped with its Heckmann convergence limH<\/sub>. \u00a0Then:<\/p>\n

Proposition.<\/strong> \u00a0For all dcpos\u00a0X<\/em>,\u00a0Y<\/em>, (X<\/em> x\u00a0Y<\/em>)H<\/sub>=X<\/i>H<\/sub>\u00a0x Y<\/em>H<\/sub>.<\/p>\n

The product on the right is product of convergence spaces. \u00a0Explicitly, for a filter\u00a0F<\/em> of subsets of\u00a0X<\/i>H<\/sub>\u00a0x Y<\/em>H<\/sub>, the convergence on the right is given by lim\u00a0F<\/em> =\u00a0limH\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>] x limH\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>]. \u00a0(The direct image\u00a0f<\/em>[F<\/em>] of a filter F<\/em>\u00a0denotes the filter {A<\/em> | f<\/em>-1<\/sup>(A<\/em>) \u2208 F<\/em>}.) \u00a0Equivalently, the proposition means that\u00a0limH\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>] x limH\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>] =\u00a0\u00a0limH\u00a0<\/sub>F<\/em>.<\/p>\n

Proof. We recall from last time<\/a> that \u222a {A<\/em>\u2193<\/sup> | A<\/em> \u2208\u00a0F<\/em>} is also the union of all ideals\u00a0I<\/em>\u00a0such that\u00a0for every element\u00a0z<\/em> of\u00a0I<\/em>,\u00a0\u2191z<\/em> is in\u00a0F<\/em>. \u00a0Hence\u00a0limH\u00a0<\/sub>F<\/em> = cl (\u222a {I<\/i> ideal of X<\/em> x\u00a0Y\u00a0<\/em>| for every\u00a0z<\/em>\u00a0\u2208\u00a0I<\/em>,\u00a0\u2191z<\/em>\u00a0\u2208 F<\/em>}).<\/p>\n

When\u00a0F<\/em> is a filter of subsets of\u00a0X<\/em> x\u00a0Y<\/em>, we must therefore examine\u00a0the ideals I<\/em>\u00a0of\u00a0X<\/em> x\u00a0Y<\/em> such that for every (x<\/em>,y<\/em>) in\u00a0I<\/em>,\u00a0\u2191(x<\/em>,y<\/em>) is in\u00a0F<\/em>. \u00a0It is an easy exercise to show that the ideals of\u00a0X<\/em> x\u00a0Y<\/em> are exactly the products of two ideals, one in\u00a0X<\/em>, the other one in\u00a0Y<\/em>. \u00a0(Or use Proposition 8.4.7 of the book<\/a>, which says that the irreducible closed subsets of a product of spaces are the products of irreducible closed subsets; Fact 8.2.49, which implies that the irreducible closed subsets of a poset in its Alexandroff topology are exactly its ideals; and Exercise 4.5.19, which says that products of posets, seen with their Alexandroff topologies, is exactly the topological product.)<\/p>\n

Therefore\u00a0limH\u00a0<\/sub>F<\/em> = cl (\u222a {I<\/i>\u00a0x J<\/em> |\u00a0I<\/em>\u00a0ideal of X<\/em>,\u00a0J<\/em> ideal of\u00a0Y<\/em>, and for all\u00a0x<\/em>\u00a0\u2208\u00a0I<\/em>, y<\/em>\u00a0\u2208\u00a0J<\/em>,\u00a0\u00a0\u2191(x<\/em>,y<\/em>)\u00a0\u2208 F<\/em>}).<\/p>\n

For every pair (x<\/em>,y<\/em>) in the union inside the closure, there is an ideal\u00a0I<\/em> of\u00a0X<\/em> containing\u00a0x<\/em>, an ideal\u00a0J<\/em> of\u00a0Y<\/em> containing y<\/em>\u00a0such that for all\u00a0x’<\/em>\u00a0\u2208\u00a0I<\/em>, y’<\/em>\u00a0\u2208\u00a0J<\/em>,\u00a0\u2191(x’<\/em>,y’<\/em>)\u00a0\u2208 F<\/em>.\u00a0Now\u00a0\u2191(x’<\/em>,y’<\/em>) =\u00a0\u2191x’<\/em>\u00a0x\u00a0\u2191y’<\/em> is included in \u2191x’<\/em>\u00a0x Y<\/em> =\u00a0\u03c01<\/sub>-1<\/sup>(\u2191x’<\/em>), so the latter is in\u00a0F<\/em>. \u00a0In turn, that means that\u00a0\u2191x’<\/em>\u00a0is in\u00a0\u03c01<\/sub>\u00a0[F<\/em>]. \u00a0Similarly,\u00a0\u2191y’<\/i>\u00a0is in\u00a0\u03c02<\/sub>\u00a0[F<\/em>]. \u00a0We obtain that\u00a0I<\/em> is an ideal containing x<\/i>\u00a0such that every\u00a0x’<\/em> in\u00a0I<\/em> satisfies\u00a0\u2191x’<\/em> \u2208 \u03c01<\/sub>\u00a0[F<\/em>], so\u00a0x<\/em> is in\u00a0limH\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>]. \u00a0Similarly\u00a0y<\/em> is in\u00a0limH\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>]. \u00a0Hence\u00a0limH\u00a0<\/sub>F<\/em> is included in\u00a0limH\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>] x limH\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>].<\/p>\n

In the converse direction, we first notice that cl(A<\/em>) x cl(B<\/em>) = cl (A<\/em> x\u00a0B<\/em>), for all subsets\u00a0A<\/em> and\u00a0B<\/em>. \u00a0Indeed,\u00a0cl(A<\/em>) x cl(B<\/em>) is closed and contains A<\/em> x B<\/em>, hence\u00a0cl (A<\/em> x\u00a0B<\/em>). \u00a0If the inclusion were strict, there would be a\u00a0point (x<\/em>,y<\/em>)\u00a0in\u00a0cl(A<\/em>) x cl(B<\/em>) but not in cl (A<\/em> x\u00a0B<\/em>). \u00a0Since the latter is closed, there would be an open rectangle U<\/em> x\u00a0V<\/em> containing\u00a0(x<\/em>,y<\/em>) and which\u00a0does not intersect cl (A<\/em> x\u00a0B<\/em>). \u00a0But\u00a0U<\/em> intersects cl(A<\/em>) at\u00a0x<\/em>, hence\u00a0U<\/em> intersects\u00a0A<\/em>, and similarly\u00a0V<\/em> intersects\u00a0B<\/em>, leading to a contradiction.<\/p>\n

Letting\u00a0A<\/em>\u00a0be the union of all ideals\u00a0I<\/em>\u00a0of X<\/em>\u00a0such that\u00a0for every element x<\/i>\u00a0of\u00a0I<\/em>,\u00a0\u2191x<\/i>\u00a0is in\u00a0\u03c01<\/sub>\u00a0[F<\/em>], and B<\/i>\u00a0be the union of all ideals J<\/i>\u00a0of Y<\/i>\u00a0such that\u00a0for every element y<\/i>\u00a0of J<\/i>,\u00a0\u2191y<\/i>\u00a0is in\u00a0\u03c02<\/sub>\u00a0[F<\/em>],\u00a0limH\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>] x limH\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>] is then\u00a0equal to cl (A<\/em> x\u00a0B<\/em>). \u00a0In order to show that\u00a0limH\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>] x limH\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>] is included in\u00a0limH\u00a0<\/sub>F<\/em>, we only have to show that\u00a0A<\/em> x\u00a0B<\/em> is\u00a0included in\u00a0limH<\/sub>\u00a0<\/sub>F<\/em>.<\/p>\n

Let (x<\/em>,y<\/em>) in\u00a0A<\/em> x\u00a0B<\/em>. \u00a0There is an ideal\u00a0I<\/em> of\u00a0X<\/em>\u00a0such that\u00a0for every element x’<\/i>\u00a0of\u00a0I<\/em>,\u00a0\u2191x’<\/i>\u00a0is in\u00a0\u03c01<\/sub>\u00a0[F<\/em>]. \u00a0There is also an ideal J<\/i>\u00a0of Y<\/i>\u00a0such that\u00a0for every element y<\/em>‘<\/i>\u00a0of J<\/i>,\u00a0\u2191y<\/em>‘<\/i>\u00a0is in\u00a0\u03c02<\/sub>\u00a0[F<\/em>]. \u00a0Then, for every\u00a0(x’<\/em>,\u00a0y’<\/em>) in\u00a0I\u00a0<\/em>x\u00a0J<\/em>,\u00a0\u2191x’<\/i>\u00a0x\u00a0Y<\/em> is in\u00a0F<\/em> and\u00a0X<\/em> x\u00a0\u2191y<\/em>‘<\/i> is in\u00a0F<\/em>. \u00a0Their intersection \u2191x’<\/i>\u00a0x\u00a0\u2191y<\/em>‘<\/i>\u00a0= \u2191(x’<\/i>,y<\/em>‘<\/i>)\u00a0is then also in\u00a0F<\/em>. \u00a0Since\u00a0I<\/em> x\u00a0J<\/em> is an ideal, (x<\/em>,y<\/em>) is in\u00a0\u222a {I<\/i>\u00a0x J<\/em> |\u00a0I<\/em>\u00a0ideal of X<\/em>,\u00a0J<\/em> ideal of\u00a0Y<\/em>, and for all\u00a0x’<\/em>\u00a0\u2208\u00a0I<\/em>, y’<\/em>\u00a0\u2208\u00a0J<\/em>,\u00a0\u00a0\u2191(x’<\/em>,y’<\/em>)\u00a0\u2208 F<\/em>}, hence in its closure limH\u00a0<\/sub>F<\/em>. \u00a0\u2610<\/p>\n

I will let you\u00a0check that this works equally well with Scott convergence, namely: not only do we have\u00a0limH<\/sub>\u00a0<\/sub>F<\/em> =\u00a0limH\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>] x limH\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>], but\u00a0limS<\/sub>\u00a0<\/sub>F<\/em> =\u00a0limS\u00a0<\/sub>\u03c01<\/sub>\u00a0[F<\/em>] x limS\u00a0<\/sub>\u03c02<\/sub>\u00a0[F<\/em>] is true as well. \u00a0The proof is the same, except that we must show that\u00a0adh(A<\/em>) x adh(B<\/em>) = adh\u00a0(A<\/em> x\u00a0B<\/em>), for all downwards-closed subsets\u00a0A<\/em> and\u00a0B<\/em> (exercise).<\/p>\n

The previous proposition also entails that the topological modification of a product\u00a0of convergence spaces is in general\u00a0different from the topological product of the topological modifications: otherwise dcpo product would coincide with topological product as well.<\/p>\n

A categorical reading of the previous proposition is that:<\/p>\n

Proposition [1].<\/strong> The functor _H<\/sub>, which maps every dcpo\u00a0X<\/em> to X<\/i>H<\/sub>, also preserves products.<\/p>\n

Proof. \u00a0The only thing that we have not proved here is that\u00a0_H<\/sub>\u00a0is indeed a functor. \u00a0For that, we should specify its action on morphisms: for every Scott-continuous map\u00a0f<\/em> from a dcpo\u00a0X<\/em> to a dcpo\u00a0Y<\/em>, we let\u00a0f<\/em>H<\/sub> be\u00a0f<\/em> itself, seen as a continuous map between convergence spaces. \u00a0Of course, we need to check that\u00a0f<\/em> is continuous in the sense of convergence spaces! \u00a0I will let you do the exercise. \u00a0This rests on elementary reasoning with ideals and filters of subsets, plus the easily proved inclusion\u00a0f<\/em>[cl(A<\/em>)]\u00a0\u2286 cl(f<\/em>[A<\/em>]). \u00a0\u2610<\/p>\n

Similarly, there is a functor\u00a0_S<\/sub>, and it too preserves products, because\u00a0f<\/em>[adh(A<\/em>)]\u00a0\u2286 adh(f<\/em>[A<\/em>]).<\/p>\n

Function spaces<\/h2>\n

Both the category\u00a0Dcpo<\/strong> of dcpos and the category\u00a0Conv<\/strong> of convergence spaces are Cartesian-closed, and one may therefore wonder whether the functor _H\u00a0<\/sub>also preserves exponential objects (function spaces, either with the pointwise ordering in Dcpo<\/strong>, or with the continuous convergence<\/a> in\u00a0Conv<\/b>).<\/p>\n

I do not know whether this\u00a0is true or not. \u00a0(I hope I have not missed anything in [1].) \u00a0For the moment, let us see what we can say.<\/p>\n

Consider two dcpos\u00a0X<\/em> and\u00a0Y<\/em>. \u00a0Write [X<\/em>\u00a0\u2192\u00a0Y<\/em>] for the dcpo of Scott-continuous maps from\u00a0X<\/em> to\u00a0Y<\/em>, and [X<\/i>H\u00a0<\/sub>\u2192 Y<\/i>H<\/sub>] for the convergence space of all continuous map from the convergence space X<\/i>H<\/sub>\u00a0to Y<\/i>H<\/sub>. \u00a0The former yields a convergence space [X<\/em>\u00a0\u2192\u00a0Y<\/em>]H<\/sub>. \u00a0Writing App for the application morphism (App sends (f<\/em>,x<\/em>) to f<\/em>(x<\/em>))\u00a0and\u00a0\u039b for currying (“evaluation”: \u039b(f<\/em>)(x<\/em>) is the map f<\/em>(x<\/em>,_)), in any of our two Cartesian-closed categories\u2014this is made possible by the fact that, set-theoretically, those acts on elements in exactly the same way, and will allow us a benign form of cheating below\u2014, we have that:<\/p>\n