{"id":1396,"date":"2018-02-26T14:14:44","date_gmt":"2018-02-26T13:14:44","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1396"},"modified":"2023-06-19T18:46:05","modified_gmt":"2023-06-19T16:46:05","slug":"meet-continuous-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1396","title":{"rendered":"Meet-continuous spaces"},"content":{"rendered":"

Meet-continuous dcpos were defined and studied by Hui\u00a0Kou,\u00a0<\/span>Ying-Ming\u00a0Liu, and Mao-Kang\u00a0Luo [1] about 14 years ago, and their importance only starts to be appreciated now. \u00a0One of the leading results in the theory of meet-continuous dcpos is that a dcpo is continuous if and only if it is quasi-continuous<\/a> and meet-continuous. \u00a0I have already talked briefly about meet-continuous dcpos here<\/a>, where I notably mentioned Weng Kin Ho, Achim Jung and Dongsheng Zhao’s new proof of that theorem through Stone duality. \u00a0Today, I would like to talk about yet another proof, mentioned by\u00a0Xiaodong Jia<\/a>\u00a0[3, Theorem 3.1.12] in his remarkable PhD thesis, and first appearing in [7, Proposition III-3.10]. \u00a0This is a much more elementary proof. \u00a0Then, I will modify it to suit my personal taste, and my exposition will progressively diverge.<\/p>\n

Meet-continuity<\/h2>\n

A lattice is\u00a0meet-continuous<\/em> if and only if the meet, or infimum operation\u00a0\u22c0 is Scott-continuous. \u00a0Meet-continuous lattices seem to have been first studied by John Isbell [2].<\/p>\n

Since a function of two arguments is Scott-continuous if and only if it is Scott-continuous in each argument separately, a lattice L<\/em>\u00a0is meet-continuous if and only if, for every\u00a0y<\/em> in\u00a0L<\/em>, the function\u00a0x<\/em>\u00a0\u21a6\u00a0x<\/em>\u00a0\u22c0\u00a0y<\/em> is Scott-continuous. \u00a0In other words, a meet-continuous complete lattice is nothing else than a frame… but that is not the direction I want to take.<\/p>\n

Since the function x<\/em>\u00a0\u21a6\u00a0x<\/em>\u00a0\u22c0\u00a0y<\/em>\u00a0is always monotonic, Scott-continuity reduces\u00a0to require that for every\u00a0y<\/em> in\u00a0L<\/em>, for every directed family\u00a0(xi<\/sub><\/em>)i \u2208 I<\/sub>, (supi \u2208 I<\/sub> xi<\/sub><\/em>) \u22c0\u00a0y<\/em> \u2264 supi \u2208 I<\/sub> (xi<\/sub><\/em>\u00a0\u22c0\u00a0y<\/em>). \u00a0(The converse inequality always holds.)<\/p>\n

Kou, Liu and Luo observed that one can give an equivalent definition that does not use the \u00a0\u22c0 operator at all. \u00a0Let us reconstruct one way of doing so. \u00a0Let\u00a0D<\/em> be a directed family (xi<\/sub><\/em>)i \u2208 I<\/sub>\u00a0of elements of\u00a0L<\/em>. \u00a0If\u00a0L<\/em> is a meet-continuous lattice, and\u00a0y<\/em>\u00a0\u2264 sup\u00a0D<\/em>, then y<\/em> =\u00a0(sup\u00a0D<\/em>) \u22c0\u00a0y<\/em> \u2264 supi \u2208 I<\/sub> (xi<\/sub><\/em>\u00a0\u22c0\u00a0y<\/em>) \u2264 sup (\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>). \u00a0That implies that\u00a0y<\/em> is in cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>), hence that \u2193y =<\/em>\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>).<\/p>\n

This is Kou, Liu and Luo’s definition of a meet-continuous dcpo: one in which\u00a0\u2193y =<\/em>\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>) for every\u00a0y<\/em>\u00a0\u2264 sup\u00a0D<\/em>, or equivalently:<\/p>\n

(meet-continuity in Kou, Liu and Luo’s sense)\u00a0for every directed family\u00a0D and every\u00a0y\u00a0\u2264 sup\u00a0D, y<\/em>\u00a0\u2208\u00a0cl\u00a0(\u2193D\u00a0\u2229\u00a0\u2193y<\/em>) .<\/p><\/blockquote>\n

We have just checked that every meet-continuous complete lattice is meet-continuous in Kou, Liu, and Luo’s sense. \u00a0Let us check the converse. \u00a0We consider a complete lattice in which y<\/em>\u00a0\u2208\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>), for every directed family\u00a0D<\/em> and every\u00a0y<\/em>\u00a0\u2264 sup\u00a0D.<\/em>\u00a0 We take a\u00a0directed family\u00a0D<\/em> =\u00a0(xi<\/sub><\/em>)i \u2208 I<\/sub>, and a point y<\/em>. \u00a0Then z<\/em> =\u00a0(supi \u2208 I<\/sub> xi<\/sub><\/em>) \u22c0\u00a0y<\/em>, which is \u2264 sup\u00a0D<\/em>, must be in\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193z<\/i>) by assumption. \u00a0We wish to show that\u00a0z<\/em> \u2264 supi \u2208 I<\/sub> (xi<\/sub><\/em>\u00a0\u22c0\u00a0y<\/em>), and for that it is enough to show that every open neighborhood\u00a0U<\/em> of\u00a0z<\/em> contains supi \u2208 I<\/sub> (xi<\/sub><\/em>\u00a0\u22c0\u00a0y<\/em>). \u00a0Since\u00a0z<\/em> is in\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193z<\/i>),\u00a0U<\/em> intersects\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193z<\/i>). \u00a0Therefore\u00a0U<\/em> also intersects\u00a0\u2193D<\/em>\u00a0\u2229\u00a0\u2193z<\/i>, say at\u00a0z’<\/em>. \u00a0Then\u00a0z’<\/em> \u2264 xi<\/sub><\/em>\u00a0for some\u00a0i<\/em>, and\u00a0z’<\/em> \u2264\u00a0z<\/em> by definition, so\u00a0z’<\/em> \u2264 xi<\/sub><\/em>\u00a0\u22c0\u00a0y<\/em>. \u00a0Since\u00a0z’<\/em> is in\u00a0U<\/em>, we conclude.<\/p>\n

Some of Xiaodong Jia’s characterizations of meet-continuity<\/h2>\n

Xiaodong’s PhD thesis sets meet-continuity at the very center of domain theory, as the title shows. \u00a0One of the things he does is give various characterizations of the notion, and we shall see some of them\u2014and also a few others. \u00a0I will not run through all of Xiaodong’s characterizations, and I will leave those that are based on\u00a0so-called forbidden structures for a later post.<\/p>\n

Before I start, let me restate a useful lemma from [4], where it appears as Lemma 3.8. \u00a0For a poset\u00a0Z<\/em>, let me write Z<\/i>\u03c3<\/sub> for\u00a0Z<\/em> with its Scott topology. \u00a0Generalizing that, given a topological space Z<\/em>, let\u00a0me write Z<\/i>\u03c3<\/sub> for\u00a0Z<\/em> with the\u00a0Scott topology of its specialization preordering. \u00a0Let us agree to call\u00a0Z<\/em> a\u00a0quasi-monotone convergence space<\/em> if the topology of Z<\/i>\u03c3<\/sub>\u00a0is finer than that of\u00a0Z<\/em>.<\/p>\n

As a vacuous example, every poset is a quasi-monotone convergence space in its Scott topology. \u00a0A much more interesting observation\u2014which I won’t make use of in this post however\u2014is that every monotone convergence space is a quasi-monotone convergence space, and every sober space is a monotone convergence space.<\/p>\n

Lemma.<\/strong> For every quasi-monotone convergence space\u00a0Z<\/em>, and every topological space Z’<\/em>,\u00a0every continuous map\u00a0f<\/em> from\u00a0Z<\/em> to\u00a0Z’<\/em> is also continuous from Z<\/i>\u03c3<\/sub>\u00a0to Z’<\/i>\u03c3<\/sub>\u00a0(i.e., Scott-continuous).<\/p>\n

Proof. Since\u00a0f<\/em> is continuous, it is monotonic, hence the only thing that we have to show is that\u00a0f<\/em>(sup\u00a0D<\/em>) \u2264 sup\u00a0f<\/em>(D<\/em>) for every directed family\u00a0D<\/em> in\u00a0Z<\/em>. \u00a0In order to do so, it suffices to show that every open neighborhood U<\/em>\u00a0of\u00a0f<\/em>(sup\u00a0D<\/em>) contains\u00a0sup\u00a0f<\/em>(D<\/em>). \u00a0Any such\u00a0U<\/em> has the property that sup\u00a0D<\/em> is in\u00a0f<\/em>-1<\/sup><\/em>(U<\/em>), which is open in\u00a0Z<\/em>, hence in\u00a0Z<\/i>\u03c3<\/sub>, since\u00a0Z<\/em> is a quasi-monotone convergence space. \u00a0Therefore some element\u00a0x<\/em> of\u00a0D<\/em> is in\u00a0f<\/em>-1<\/sup><\/em>(U<\/em>), hence\u00a0f<\/em>(x<\/em>) is in U<\/em>. \u00a0Then\u00a0f<\/em>(sup\u00a0D<\/em>), which is even larger, is also in\u00a0U<\/em>. \u00a0\u00a0\u2610<\/p>\n

Given a topological space X<\/em>, we consider the space\u00a0H<\/strong>(X<\/em>) of closed subsets of\u00a0X<\/em>. I have already mentioned that this is the Hoare hyperspace<\/a> of\u00a0X<\/em>. \u00a0\u00a0Its topology\u2014the lower Vietoris topology\u2014has a subbase consisting of sets \u25caU<\/em>, U<\/em> open in X<\/em>, where by definition \u25caU<\/em> is the set of closed sets F\u00a0<\/em>of X that intersect U<\/em>. \u00a0The specialization ordering of H<\/strong>(X<\/em>) is inclusion, and there is a continuous map\u00a0\u03b7 :\u00a0x<\/em>\u00a0\u2208\u00a0X<\/em>\u00a0\u21a6\u00a0\u2193x<\/em>\u00a0\u2208\u00a0H<\/strong>(X<\/em>).<\/p>\n

Under inclusion H<\/strong>(X<\/em>) is a complete lattice, and it is traditional in domain theory to give it the Scott topology. \u00a0In general the Scott topology is finer than the lower Vietoris topology, and it coincides with it when\u00a0X<\/em> is a c-space (see last proposition in the above mentioned post<\/a>). Since we do not assume\u00a0X<\/em> to be a c-space, one should really be careful and distinguish\u00a0H<\/strong>(X<\/em>) from H<\/strong>(X<\/em>)\u03c3<\/sub><\/em>.<\/p>\n

Proposition.<\/strong> Given a dcpo\u00a0X<\/em>, the following are equivalent:<\/p>\n

    \n
  1. X<\/em> is meet-continuous;<\/li>\n
  2. for every\u00a0y<\/em> in\u00a0X<\/em>, the map \u03b7y<\/sub><\/em> :\u00a0\u00a0x<\/em>\u00a0\u2208\u00a0X<\/em>\u00a0\u21a6\u00a0\u2193x<\/em>\u00a0\u2229\u00a0\u2193y<\/em> is continuous from\u00a0X\u03c3<\/sub><\/em> to H<\/strong>(X<\/em>);<\/li>\n
  3. for every\u00a0y<\/em> in\u00a0X<\/em>, the map \u03b7y<\/sub><\/em> :\u00a0\u00a0x<\/em>\u00a0\u2208\u00a0X<\/em>\u00a0\u21a6\u00a0\u2193x<\/em>\u00a0\u2229\u00a0\u2193y<\/em> is (Scott-)continuous from X\u03c3<\/sub><\/em>\u00a0to H<\/strong>(X<\/em>)\u03c3<\/sub><\/em>;<\/li>\n
  4. for every y<\/i>\u00a0in\u00a0X<\/em>, for every (Scott-)open subset\u00a0U<\/em> of X<\/i>,\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>) is (Scott-)open;<\/li>\n
  5. H<\/strong>(X<\/em>) is a meet-continuous complete lattice;<\/li>\n
  6. for all downwards closed subsets\u00a0A<\/em> and\u00a0B<\/em>, cl (A<\/em>)\u00a0\u2229 cl (B<\/em>) = cl (A<\/em>\u00a0\u2229\u00a0B<\/em>);<\/li>\n
  7. for all upwards closed subset\u00a0A<\/em> and\u00a0B<\/em>, int (A<\/em>)\u00a0\u222a int (B<\/em>) = int (A<\/em>\u00a0\u222a\u00a0B<\/em>).<\/li>\n<\/ol>\n

    The equivalence between 1 and 3 is Proposition 3.1.5 of [3]. \u00a0The equivalence between 1 and 5 is Proposition 3.2.9. \u00a0Item 4 is mentioned in Theorem 3.2.1.<\/p>\n

    Proof.<\/em>\u00a0 1\u21d22. \u00a0 Take an arbitrary point y<\/em> in\u00a0X<\/em>, an arbitrary open subset\u00a0U<\/em> of\u00a0X<\/em>. \u00a0We wish to show that \u03b7y<\/sub><\/em>-1<\/sup>(\u25caU<\/em>) is Scott-open. \u00a0It is easy to see that it is upwards-closed. \u00a0We take\u00a0an\u00a0arbitrary directed family\u00a0D<\/em>, and we assume that sup D<\/em> is in\u00a0\u03b7y<\/sub><\/em>-1<\/sup>(\u25caU<\/em>). \u00a0Hence \u2193sup\u00a0D<\/em>\u00a0\u2229\u00a0\u2193y<\/em> intersects U<\/em>, say at\u00a0x<\/em>. \u00a0Since\u00a0x<\/em> \u2264 sup\u00a0D<\/em> and 1 holds,\u00a0x<\/em> is in\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193x<\/i>). \u00a0It follows that the latter intersects\u00a0U<\/em>, hence also\u00a0\u2193D<\/em>\u00a0\u2229\u00a0\u2193x<\/i> intersects\u00a0U<\/em>. \u00a0Pick a point in that intersection: it is in\u00a0U<\/em>, below some point\u00a0z<\/em> of\u00a0D<\/em>, and below\u00a0x<\/em>. \u00a0Hence\u00a0\u2193z<\/em>\u00a0\u2229\u00a0\u2193y<\/em> intersects\u00a0U<\/em>, showing that\u00a0z<\/em> (a point of\u00a0D<\/em>) is in\u00a0\u03b7y<\/sub><\/em>-1<\/sup>(\u25caU<\/em>).<\/p>\n

    2\u21d23. \u00a0X\u03c3\u00a0<\/sub><\/em>is a quasi-monotone convergence space: then apply the previous Lemma.<\/p>\n

    3\u21d24. \u00a0Assuming 3, item 2 holds\u00a0because the Scott topology on H<\/strong>(X<\/em>) is finer than the Vietoris topology. \u00a0Hence for every y<\/i>\u00a0in\u00a0X<\/em>,\u00a0\u03b7y<\/sub><\/em>-1<\/sup>(\u25caU<\/em>) is Scott-open. \u00a0However,\u00a0\u03b7y<\/sub><\/em>-1<\/sup>(\u25caU<\/em>) is\u00a0the set of points x<\/i>\u00a0such that\u00a0\u2193x<\/em>\u00a0\u2229\u00a0\u2193y<\/em> intersects\u00a0U<\/em>, equivalently the set of points above some point in\u00a0\u2193y<\/em>\u00a0\u2229\u00a0U<\/em>,\u00a0and that is\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>).<\/p>\n

    4\u21d21. \u00a0Assume y<\/em>\u00a0\u2264 sup\u00a0D<\/em>, where D<\/em> is some directed family of elements of\u00a0X<\/em>,\u00a0and let\u00a0U<\/em> be\u00a0the complement of\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>). \u00a0We wish to show that y<\/em> is in\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>). \u00a0If that were not the case, then\u00a0y<\/em> would be in\u00a0U<\/em>, hence also in\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>). \u00a0The latter is Scott-open by 4, and since\u00a0y<\/em>\u00a0\u2264 sup\u00a0D<\/em>, some element z<\/i>\u00a0of\u00a0D<\/em> must be in\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>). \u00a0Hence there is a point\u00a0x<\/em> in\u00a0U<\/em> such that\u00a0x<\/em>\u2264y and\u00a0x\u2264z<\/em>. \u00a0But now x<\/em> is in\u00a0\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>, hence in\u00a0cl\u00a0(\u2193D<\/em>\u00a0\u2229\u00a0\u2193y<\/em>), and also in\u00a0U<\/em>: contradiction.<\/p>\n

    It remains to show that any of the equivalent items\u00a01\u20144 is equivalent with 5, 6, and 7. \u00a0We show\u00a04\u21d25\u21d26\u21d27\u21d24.<\/p>\n

    4\u21d25. \u00a0We only have to show that intersection is Scott-continuous on H<\/strong>(X<\/em>). \u00a0Let\u00a0C<\/em> be a closed subset of\u00a0X<\/em>, and\u00a0(Ci<\/sub><\/em>)i \u2208 I<\/sub>\u00a0be a directed family of closed subsets of\u00a0X<\/em>, with supremum\u00a0C’<\/em>. \u00a0We must show that\u00a0C<\/em>\u00a0\u2229 C’<\/em>\u00a0\u2286\u00a0supi<\/sub>\u00a0<\/sub><\/em>(C <\/em>\u2229 Ci<\/sub><\/em>), where sup denotes closure of union. \u00a0(The converse inclusion is immediate since intersection is monotonic.) \u00a0In order to do so, we take an arbitrary open set\u00a0U<\/em> that intersects C<\/em>\u00a0\u2229 C’<\/em>, and we show that it intersects supi<\/sub>\u00a0<\/sub><\/em>(C <\/em>\u2229 Ci<\/sub><\/em>). \u00a0Since\u00a0U<\/em>\u00a0intersects C<\/em>\u00a0\u2229 C’<\/em>, there is a point y<\/i>\u00a0in\u00a0U<\/em> that is in\u00a0C<\/em> and in\u00a0C’<\/em>. \u00a0By item 4,\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>) is open, and intersects\u00a0C’<\/em> at\u00a0y<\/em>. \u00a0Since it intersects\u00a0C’<\/em>\u00a0= cl (\u222ai<\/sub>\u00a0Ci<\/sub><\/em>) and it is open, it intersects \u222ai<\/sub>\u00a0Ci<\/sub><\/em>, hence some Ci<\/sub><\/em>, say at\u00a0z<\/em>. \u00a0This z<\/i>\u00a0is in\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>), hence is above some point\u00a0x<\/em> of\u00a0U<\/em> such that\u00a0x<\/em>\u2264y<\/em>. \u00a0Since Ci<\/sub><\/em>\u00a0is downwards closed,\u00a0x<\/em> is in Ci<\/sub><\/em>, and since\u00a0C<\/em> is downwards closed and contains\u00a0y<\/em>,\u00a0x<\/em> is also in\u00a0C<\/em>. \u00a0This shows that\u00a0U<\/em> intersects C <\/em>\u2229 Ci<\/sub><\/em>\u00a0(at x<\/em>),\u00a0hence also the larger set\u00a0supi<\/sub>\u00a0<\/sub><\/em>(C <\/em>\u2229 Ci<\/sub><\/em>).<\/p>\n

    5\u21d26. \u00a0The inclusion\u00a0cl (A<\/em>\u00a0\u2229\u00a0B<\/em>)\u00a0\u2286\u00a0cl (A<\/em>)\u00a0\u2229 cl (B<\/em>) is always true. \u00a0In the converse direction, cl (A<\/em>) is the supremum in\u00a0H<\/strong>(X<\/em>) of the directed family of elements of the form\u00a0\u2193F<\/em>, where\u00a0F<\/em>\u00a0ranges over the finite subsets of\u00a0A<\/em>, and cl (B<\/em>) is the supremum of the directed family of elements of the form\u00a0\u2193G<\/em>, where G<\/i> ranges over the finite subsets of\u00a0B<\/i>. \u00a0By item 5,\u00a0cl (A<\/em>)\u00a0\u2229 cl (B<\/em>) is then equal to the supremum of the family of all subsets of the form \u2193F<\/em>\u00a0\u2229\u2193G<\/em>, where F<\/em> and\u00a0G<\/em> are as above, and those subsets are all included in A<\/em>\u00a0\u2229\u00a0B<\/em>, hence in cl (A<\/em>\u00a0\u2229\u00a0B<\/em>).<\/p>\n

    6\u21d27. \u00a0Follows directly by complementation, and the realization that the complements of downwards closed sets are exactly the upwards closed sets.<\/p>\n

    7\u21d24. \u00a0For every open subset\u00a0U<\/em>,\u00a0U<\/em> is included in\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>) \u222a V<\/em>, where V<\/em> is the complement of\u00a0\u2193y<\/em>, as one checks by realizing that every element of\u00a0U<\/em> is either below\u00a0y<\/em> or not. \u00a0Hence\u00a0U<\/em> is included in the interior of\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>) \u222a V<\/em>, which is equal to int (\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>)) \u222a int (V<\/em>) by item 7. \u00a0Now\u00a0V<\/em> is open, so int (V<\/em>) =\u00a0V<\/em>. \u00a0From\u00a0U<\/em>\u00a0\u2286\u00a0int (\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>)) \u222a\u00a0V<\/em>, we obtain\u00a0U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>\u00a0\u2286\u00a0int (\u2191(U\u00a0<\/em>\u2229\u00a0\u2193y<\/em>)), hence\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>) \u2286\u00a0int (\u2191(U\u00a0<\/em>\u2229\u00a0\u2193y<\/em>)), and that shows that\u00a0\u2191(U<\/em>\u00a0\u2229\u00a0\u2193y<\/em>) is open. \u00a0 \u00a0\u2610<\/p>\n

    That proposition suggests several ways of generalizing the notion of meet-continuous dcpo to all topological spaces. \u00a0Maybe you have noticed that the proofs I gave of the implications\u00a04\u21d25\u21d26\u21d27\u21d24 work in any topological space. \u00a0Hence I would like to suggest the following.<\/p>\n

    Definition (meet-continuous topological space).<\/strong> \u00a0A topological space is\u00a0meet-continuous<\/em> if and only if any of the following equivalent conditions hold:<\/p>\n