{"id":7144,"date":"2023-09-20T19:39:30","date_gmt":"2023-09-20T17:39:30","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=7144"},"modified":"2023-09-20T19:41:01","modified_gmt":"2023-09-20T17:41:01","slug":"scotts-formula","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=7144","title":{"rendered":"Scott’s formula"},"content":{"rendered":"\n

There is a famous formula in domain theory, called Scott’s formula<\/em>. If B<\/em> is a basis of a continuous poset X<\/em>, and f<\/em> is a monotonic function from B<\/em> to some dcpo Y<\/em>, then one can define a new function f’<\/em> from the whole of X<\/em> to Y<\/em> by:<\/p>\n\n\n\n

f’<\/em>(x<\/em>) \u225d supb<\/em> \u2208 B<\/em>, b<\/em> \u226a x<\/em><\/sub> f<\/em>(b<\/em>),<\/p>\n\n\n\n

and then f’<\/em> will automatically be a continuous function from X<\/em> to Y<\/em>. There is no reason why it would coincide with f<\/em> on B<\/em>, but it is the best continuous approximation to f<\/em> on B<\/em>, in the sense that f’<\/em> is below f<\/em> on B<\/em> (f’<\/em>(b<\/em>)\u2264f<\/em>(b<\/em>) for every b<\/em> \u2208 B<\/em>) and that it is the largest one with that property (for every continuous function g<\/em> below f<\/em> on B<\/em>, g<\/em>\u2264f’<\/em>).<\/p>\n\n\n\n

This is mentioned as Proposition 5.1.60 in the book<\/a>, although you should definitely read the errata<\/a> page (important blooper #13).<\/p>\n\n\n\n

I would like to explore a few ways that this formula can be extended, in what cases f’<\/em> coincides with f<\/em> on B<\/em> (i.e., in what cases f’<\/em> is a continuous extension<\/em> of f<\/em>), and finally in what cases any algebraic laws satisfied by f<\/em> will still be satisfied by f’<\/em>.<\/p>\n\n\n\n

c-spaces<\/h2>\n\n\n\n

We can first extend Scott’s formula to the case where X<\/em> is a c-space<\/em>, not just a continuous poset. A c-space is a space X<\/em> in which, for every point x<\/em>, for every open neighborhood U<\/em> of x<\/em>, there is a point y<\/em> in U<\/em> such that x<\/em> is in the interior of \u2191y<\/em>.<\/p>\n\n\n\n

It will be practical to write y<\/em> \u226a x<\/em> instead of “x<\/em> is in the interior of \u2191y<\/em>“, and \u219fy<\/em> for the interior of \u2191y<\/em> (not just in c-spaces, but in any topological space). You might be concerned about a possible conflict of notations with the way-below relation on posets. But, in the case where X<\/em> is a continuous poset, with its Scott topology, the usual way-below relation \u226a is exactly given by y<\/em> \u226a x<\/em> if and only if x<\/em> is in the interior of \u2191y<\/em>, or equivalently, the interior of \u2191y<\/em> is exactly \u219fy<\/em>, defined as the set of elements z<\/em> such that y<\/em> is way-below z<\/em> (this is Proposition 5.1.35 of the book<\/a>); and I will never consider the way-below relation on posets that are not continuous in this post.<\/p>\n\n\n\n

Now let me define a basis<\/em> of a topological space X<\/em> as a collection B<\/em> of points of X<\/em> such that for every point x<\/em> in X<\/em>, for every open neighborhood U<\/em> of x<\/em>, there is an element b<\/em> of B<\/em> \u2229 U<\/em> such that b<\/em> \u226a x<\/em> (i.e., such that x<\/em> is in the interior of \u2191b<\/em>). Any space with a basis must be c-space, and conversely, and c-space X<\/em> has a basis, namely X<\/em> itself. When X<\/em> is a continuous poset with its Scott topology, we retrieve the usual definition of a basis.<\/p>\n\n\n\n

Scott’s formula has an easy generalization to c-spaces and their bases, as we now see. Note that X<\/em> need not be sober, in particular; if X<\/em> were sober, then it would be a continuous dcpo with its Scott topology (Proposition 8.3.36 of the book<\/a>). There is no harm in replacing Y<\/em>, with its Scott topology, with a monotone convergence space either. All spaces are (pre)ordered with their specialization preorderings; this is what allows us to make sense of the adjective “monotonic” below.<\/p>\n\n\n\n

Theorem.<\/strong> Let B<\/em> be a basis of a c-space X<\/em>, and f<\/em> be a monotonic function from B<\/em> to some monotone convergence space Y<\/em>. For every x<\/em> in X<\/em>, let:<\/p>\n\n\n\n

f’<\/em>(x<\/em>) \u225d supb<\/em> \u2208 B<\/em>, b<\/em> \u226a x<\/em><\/sub> f<\/em>(b<\/em>)
(Scott’s formula).<\/p>\n\n\n\n

Then f’<\/em> is a continuous map from X<\/em> to Y<\/em>, and is the largest continuous map on X<\/em> that is below f<\/em> on B<\/em>.<\/p>\n\n\n\n

Proof.<\/em> For any given point x<\/em>, let Fx<\/sub><\/em> denote the family of elements b<\/em> \u2208 B<\/em> such that b<\/em> \u226a x<\/em>. We claim that Fx<\/sub><\/em> is directed. Once we have proved this, the supremum that defines f’<\/em>(x<\/em>) will be a supremum of a directed family, hence will be well-defined, since Y<\/em> is a monotone convergence space, in particular a dcpo in its specialization ordering. Now every open neighborhood U<\/em> of x<\/em> contains an element of Fx<\/sub><\/em>, since B<\/em> is a basis of X<\/em>. By taking U<\/em> \u225d X<\/em>, Fx<\/sub><\/em> is non-empty, and for any two elements b<\/em>1<\/sub> and b<\/em>2<\/sub> of Fx<\/sub><\/em>, there is an element b<\/em> of Fx<\/sub><\/em> in int(\u2191 b<\/em>1<\/sub>) \u2229 int(\u2191b<\/em>2<\/sub>); in particular, b<\/em>1<\/sub>, b<\/em>2<\/sub> \u226a b<\/em>. Hence Fx<\/sub><\/em> is directed, as promised.<\/p>\n\n\n\n

We claim that f’<\/em> is continuous. For every open subset V<\/em> of Y<\/em>, for every x<\/em> \u2208 X<\/em>, f’<\/em>(x<\/em>) \u2208 V<\/em> if and only if supb<\/em> \u2208 B<\/em>, b<\/em> \u226a x<\/em><\/sub> f<\/em>(b<\/em>) is in V<\/em>. Since Y<\/em> is a monotone convergence space, V<\/em> is Scott-open, so supb<\/em> \u2208 B<\/em>, b<\/em> \u226a x<\/em><\/sub> f<\/em>(b<\/em>) \u2208 V<\/em> implies that f<\/em>(b<\/em>) \u2208 V<\/em> for some b<\/em> \u2208 B<\/em> such that b<\/em> \u226a x<\/em>. Conversely, if f<\/em>(b<\/em>) \u2208 V<\/em> for some b<\/em> \u2208 B<\/em> such that b<\/em> \u226a x<\/em>, then f’<\/em>(x<\/em>), which is even larger, is in V<\/em>. Hence f’<\/em>\u22121<\/sup>(V<\/em>) is the union of the sets \u219fb<\/em>, where b<\/em> ranges over the elements of B<\/em> such that f<\/em>(b<\/em>) \u2208 V<\/em>. In other words, f’<\/em>\u22121<\/sup>(V<\/em>) = \u222ab<\/em> \u2208 f<\/em>\u22121<\/sup> (V)<\/sub> int(\u2191b<\/em>). Since that is open, f’<\/em> is continuous.<\/p>\n\n\n\n

For every x<\/em> \u2208 B<\/em>, we have that every b<\/em> \u2208 B<\/em> such that b<\/em> \u226a x<\/em> (i.e., x<\/em> \u2208 int(\u2191b<\/em>)) is such that x<\/em> is in \u2191b<\/em>, namely b<\/em> \u2264 x<\/em>; so f<\/em>(b<\/em>) \u2264 f<\/em>(x<\/em>) since f<\/em> is monotonic. Taking suprema, we obtain that f’<\/em>(x<\/em>)\u2264f<\/em>(x<\/em>). Hence f’<\/em> is below f<\/em> on B<\/em>.<\/p>\n\n\n\n

Finally, we consider any continuous map g<\/em> from X<\/em> to Y<\/em> that is below f<\/em> on B<\/em>. For every point x<\/em> of X<\/em>, for every open neighborhood V<\/em> of g<\/em>(x<\/em>), g<\/em>\u22121<\/sup>(V<\/em>) is an open neighborhood of x<\/em>, hence contains some element b<\/em> of B<\/em> such that b<\/em> \u226a x<\/em>, since B<\/em> is a basis of X<\/em>. Since g<\/em>(b<\/em>) \u2264 f<\/em>(b<\/em>) and g<\/em>(b<\/em>) \u2208 V<\/em>, f<\/em>(b<\/em>) is also in V<\/em>, and therefore f’<\/em>(x<\/em>) \u2265 f<\/em>(b<\/em>) is in V<\/em>. We have shown that every open neighborhood V<\/em> of g<\/em>(x<\/em>) contains f’<\/em>(x<\/em>), so g<\/em>(x<\/em>) \u2264 f’<\/em>(x<\/em>), showing that f’<\/em> is the largest continuous map below f<\/em> on B<\/em>. \u2610<\/p>\n\n\n\n

Continuous extensions<\/h2>\n\n\n\n

The view I have just given of Scott’s formula on c-spaces gives us a complete characterization of when f’<\/em> is an extension of f<\/em>.<\/p>\n\n\n\n

Proposition.<\/strong> Let B<\/em> be a basis of a c-space X<\/em>, f<\/em> be a monotonic map from B<\/em> to a monotone convergence space Y<\/em>, and let f’<\/em> be defined by Scott’s formula. Then f’<\/em> extends f<\/em> if and only if f<\/em> is continuous from B<\/em> (with the subspace topology from X<\/em>) to Y<\/em>.<\/p>\n\n\n\n

Proof.<\/em> We have seen that f’<\/em> is continuous from X<\/em> to Y<\/em>. Therefore its restriction to the subspace B<\/em> is continuous. In particular, if f’<\/em> coincides with f<\/em> on B<\/em>, f<\/em> must be continuous from B<\/em> to Y<\/em>.<\/p>\n\n\n\n

Conversely, let us assume that f<\/em> is continuous from B<\/em> to Y<\/em>. For every open subset V<\/em> of Y<\/em>, f<\/em>\u22121<\/sup>(V<\/em>) is open, hence equal to U<\/em> \u2229 B<\/em> for some open subset U<\/em> of X<\/em>. For every x<\/em> \u2208 f<\/em>\u22121<\/sup>(V<\/em>)= U<\/em> \u2229 B<\/em>, by definition of B<\/em> there is an element b<\/em> \u2208 B<\/em> such that b<\/em> \u226a x<\/em> and b<\/em> \u2208 U<\/em>. Hence b<\/em> is in f<\/em> \u22121<\/sup>(V<\/em>). Equivalently, f<\/em>(b<\/em>) \u2208 V<\/em>, and this shows that f’<\/em>(x<\/em>) \u2265 f<\/em>(b<\/em>) is in V<\/em>. We have shown that for every x<\/em> \u2208 X<\/em>, every open neighborhood V<\/em> of f<\/em>(x<\/em>) also contains f’<\/em>(x<\/em>), so f<\/em>(x<\/em>) \u2264 f’<\/em>(x<\/em>). Since f’<\/em> is below f<\/em> on B<\/em>, we conclude that f<\/em> and f’<\/em> coincide on B<\/em>. \u2610<\/p>\n\n\n\n

When X<\/em> is a continuous poset with basis B<\/em>, and Y<\/em> is a dcpo, the condition that f<\/em> be continuous from B<\/em>, with the subspace topology from X<\/em>, to Y<\/em>, is equivalent to the fact that f<\/em> be relatively Scott-continuous on B<\/em>, which I will define next. There is a pretty subtle point here, as any Scott-continuous map from B<\/em> to Y<\/em> will be relatively Scott-continuous on B<\/em>, but the converse may fail.<\/p>\n\n\n\n

Let me call X<\/em>–supremum<\/em> of a family D<\/em> of points of X<\/em> the least upper bound of D<\/em> in X<\/em>, if it exists. Let me call B<\/em>–supremum<\/em> of a family D<\/em> of points of B<\/em> the least upper bound of D<\/em> in B<\/em>, if it exists. Now, given any family D<\/em> of points of B<\/em>,<\/p>\n\n\n\n

Next, we examine when f’<\/em> is an extension of f<\/em>, namely when f’<\/em> coincides with f<\/em> on B<\/em>. This is dealt with in Corollary 5.1.61 of the book<\/a>; as with Proposition 5.1.60, you need Y<\/em> to be a dcpo, not just a bdcpo (see the errata<\/a> page, important blooper #13; you may also keep Y<\/em> as a bdcpo, and then you need to require B<\/em> to be cofinal in X<\/em>).<\/p>\n\n\n\n