{"id":6923,"date":"2023-06-20T15:34:05","date_gmt":"2023-06-20T13:34:05","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6923"},"modified":"2023-06-20T15:47:46","modified_gmt":"2023-06-20T13:47:46","slug":"the-banaschewski-lawson-ershov-observation-on-separate-vs-joint-continuity","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6923","title":{"rendered":"The Banaschewski-Lawson-Ershov observation on separate vs. joint continuity"},"content":{"rendered":"\n

I was planning to continue the subject of exponentiable locales<\/a>, but that grew out of hand. Instead, let me touch a lighter subject today. If you thought that my posts were more and more complex each time, that should be a relief to you. If you wish to complain that I am not giving you enough thought for food, or that I am becoming lazy, please consider that I posted twice last month, and that the post on the seminar on continuity in semilattices<\/a> points to 98 papers for you to peruse, no less! 😀<\/p>\n\n\n\n

Separate and joint continuity<\/h2>\n\n\n\n

The question we will address today is in the following setting. We consider a function f<\/em> : X<\/em> \u00d7 Y<\/em> \u2192 Z<\/em>, where X<\/em>, Y<\/em>, Z<\/em> are topological spaces, and we know that f<\/em> is separately<\/em> continuous; we would like to prove that f<\/em> is jointly<\/em> continuous. When is that true?<\/p>\n\n\n\n

You have probably been warned that there are functions in two arguments in nature that are separately continuous but not jointly continuous, and Exercise 4.5.11 in the book<\/a> is a classical counterexample. However, there are cases where separate continuity implies<\/em> (and therefore, is equivalent to) joint continuity.<\/p>\n\n\n\n

The Ershov observation<\/h2>\n\n\n\n

I first learned of the following result, due to Yuri Ershov [1, Proposition 2] from Klaus Keimel, who used it to show that if the scalar multiplication on a cone C<\/em> (whatever that is; this is not important), which happens to be a map from R<\/strong>+<\/sub> \u00d7 C<\/em> to C<\/em>, is separately continuous, then it is automatically jointly continuous [2]. Here R<\/strong>+<\/sub> has to have the Scott topology of its usual ordering \u2264.<\/p>\n\n\n\n

For some time, I called the following the Ershov observation<\/em>. 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 the upward closure \u2191y<\/em> of y<\/em>; the sober c-spaces are exactly the continuous dcpos in their Scott topology. This applies in the case of scalar multiplication cited above because R<\/strong>+<\/sub>, with its usual Scott topology (not<\/em> its metric topology) is a c-space; in fact, a continuous poset.<\/p>\n\n\n\n

Proposition (Ershov).<\/strong> Let X<\/em> be a c-space, and Y<\/em>, Z<\/em> be arbitrary topological spaces. A function f<\/em> : X<\/em> \u00d7 Y<\/em> \u2192 Z<\/em> is separately continuous if and only if it is jointly continuous.<\/p>\n\n\n\n

Let me give a proof of this fact. This is not hard. However, please be warned that we will give another theorem below which subsumes this one.<\/p>\n\n\n\n

Proof.<\/em> Let us assume that f<\/em> is separately continuous. Let W<\/em> be an open subset of Z<\/em>, and (x<\/em>, y<\/em>) \u2208 f<\/em> \u22121<\/sup>(W<\/em>). Since f<\/em> is separately continuous, f<\/em> (_, y<\/em>) is continuous, and therefore f<\/em> (_, y<\/em>)\u22121<\/sup>(W<\/em>) is an open neighborhood of x<\/em>. We use the fact that X<\/em> is a c-space and we obtain a point x’<\/em> \u2208 f<\/em> (_, y<\/em>)\u22121<\/sup>(W<\/em>) such that x<\/em> \u2208 int(\u2191x’<\/em>).<\/p>\n\n\n\n

Let us use the fact that f<\/em> is separately continuous a second time, but this time on the other argument: f<\/em> (x’<\/em>, _) is continuous; so f<\/em> (x’<\/em>, _)\u22121<\/sup>(W<\/em>) is open. We had built x<\/em>‘ so that x’<\/em> \u2208 f<\/em> (_, y<\/em>)\u22121<\/sup>(W<\/em>), so f<\/em> (x’<\/em>, y<\/em>) is in W<\/em>, and hence y<\/em> is in f<\/em> (x’<\/em>, _)\u22121<\/sup>(W<\/em>).<\/p>\n\n\n\n

Then int(\u2191x’<\/em>) \u00d7 V<\/em> is an open neighborhood of (x<\/em>, y<\/em>) in the product topology, and it remains to show that it is included in f<\/em>\u22121<\/sup>(W<\/em>). For every pair (x”<\/em>, y”<\/em>) \u2208 int(\u2191x’<\/em>) \u00d7 V<\/em>, f<\/em> (x”<\/em>, y”<\/em>) is larger than or equal to f<\/em> (x’<\/em>, y”<\/em>) since x’<\/em> \u2264 x”<\/em> and since every continuous map is monotonic. By construction, f<\/em> (x’<\/em>, y”<\/em>) is in W<\/em>, and therefore f<\/em> (x”<\/em>, y”<\/em>) is also in W<\/em>, since W<\/em> is upwards-closed. \u2610<\/p>\n\n\n\n

The Banaschewski-Lawson observation, part I<\/h2>\n\n\n\n

A few years later, I realized that some people (Zhenchao Lyu, Xiaolin Xie, and Hui Kou) knew better than I did: Jimmie Lawson had proved a rather sweeping generalization of the Ershov observation… twelve years earlier [3, Theorem 2]. I have already mentioned this in my post on the ISDT’22 conference<\/a>. It is time I gave the proof.<\/p>\n\n\n\n

Oh, if you read the paragraph before Theorem 2 in [3], you will realize that Jimmie Lawson says that most of that Theorem is due to Bernard Banaschewski. And indeed, the following is part of Proposition 6 of [4], which predates [3] by five more years. But then, as J. Lawson notices, Banaschewski’s Corollary 2 to his Proposition 3 is incorrect, and is used at some point… it is always so complicated to trace back the history of a result!<\/p>\n\n\n\n

Proposition (Banaschewski-Lawson, part I).<\/strong> Let X<\/em> be a locally finitary compact space, and Y<\/em>, Z<\/em> be arbitrary topological spaces. A function f<\/em> : X<\/em> \u00d7 Y<\/em> \u2192 Z<\/em> is separately continuous if and only if it is jointly continuous.<\/p>\n\n\n\n

This is exactly the same statement as with Ershov’s observation, except that X<\/em> is only assumed to be locally finitary compact<\/em>. In other words, for every point x<\/em> of X<\/em>, for every open neighborhood U<\/em> of x<\/em>, there is a finitary compact<\/em> set \u2191E<\/em> (where E<\/em>, by definition, is finite) such that x<\/em> \u2208 int(\u2191E<\/em>) \u2286 \u2191E<\/em> \u2286 U<\/em>. This is a strong form of local compactness, where the interpolating compact set Q<\/em> (i.e., such that x<\/em> \u2208 int(Q<\/em>) \u2286 \u2191Q<\/em> \u2286 U<\/em>) has to be the upward closure of finitely many points. C-spaces are an even stronger form of local compactness, where E<\/em> is required to contain exactly one point.<\/p>\n\n\n\n

Proof.<\/em> The proof is exactly as with Ershov’s observation, changing only the details that must change. <\/p>\n\n\n\n

Let us assume that f<\/em> is separately continuous. Let W<\/em> be an open subset of Z<\/em>, and (x<\/em>, y<\/em>) \u2208 f<\/em> \u22121<\/sup>(W<\/em>). Since f<\/em> is separately continuous, f<\/em> (_, y<\/em>) is continuous, and therefore f<\/em> (_, y<\/em>)\u22121<\/sup>(W<\/em>) is an open neighborhood of x<\/em>. We use the fact that X<\/em> is locally finitary compact and we obtain a finite subset A<\/em> of f<\/em> (_, y<\/em>)\u22121<\/sup>(W<\/em>) such that x<\/em> \u2208 int(\u2191A<\/em>)\u2014instead of just a single point x’<\/em>, as in the proof of Ershov’s observation.<\/p>\n\n\n\n

Let us use the fact that f<\/em> is separately continuous a second time, but this time on the other argument. For every x’<\/em> \u2208 A<\/em>, f<\/em> (x’<\/em>, _) is continuous, so f<\/em> (x’<\/em>, _)\u22121<\/sup>(W<\/em>) is open. By definition of A<\/em>, x’<\/em> is in f<\/em> (_, y<\/em>)\u22121<\/sup>(W<\/em>), so f<\/em> (x’<\/em>, y<\/em>) is in W<\/em>, and hence y<\/em> is in f<\/em> (x’<\/em>, _)\u22121<\/sup>(W<\/em>).<\/p>\n\n\n\n

Let V<\/em> be the intersection of all the open sets f<\/em> (x’<\/em>, _)\u22121<\/sup>(W<\/em>), when x’<\/em> ranges over A<\/em>. Since A<\/em> is finite, V<\/em> is open. Also, y<\/em> is in V<\/em>.<\/p>\n\n\n\n

Then int(\u2191A<\/em>) \u00d7 V<\/em> is an open neighborhood of (x<\/em>, y<\/em>) in the product topology, and it remains to show that it is included in f<\/em>\u22121<\/sup>(W<\/em>). For every pair (x”<\/em>, y”<\/em>) \u2208 int(\u2191A<\/em>) \u00d7 V<\/em>, x”<\/em> is in particular in \u2191A<\/em>, so there is a point x’<\/em> in A<\/em> such that x’<\/em>\u2264x”<\/em>. Since every continuous map is monotonic, it follows that f<\/em> (x”<\/em>, y”<\/em>) is larger than or equal to f<\/em> (x’<\/em>, y”<\/em>). By construction, y”<\/em> is in V<\/em>, hence in f<\/em> (x’<\/em>, _)\u22121<\/sup>(W<\/em>), so f<\/em> (x’<\/em>, y”<\/em>) is in W<\/em>. Therefore f<\/em> (x”<\/em>, y”<\/em>) is also in W<\/em>, since W<\/em> is upwards-closed. \u2610<\/p>\n\n\n\n

The Banaschewski-Lawson observation, part II<\/h2>\n\n\n\n

What makes the Banaschewski-Lawson observation more remarkable is that the condition is not only sufficient, it is necessary<\/em>. The following is the equivalence between items (6) and (8) in J. Lawson’s Theorem 2 in [3].<\/p>\n\n\n\n

Theorem (Banaschewski-Lawson).<\/strong> The class C<\/strong> of topological spaces X<\/em> such that for all topological spaces Y<\/em> and Z<\/em>, every separately continuous function f<\/em> : X<\/em> \u00d7 Y<\/em> \u2192 Z<\/em> is jointly continuous is exactly<\/em> the class of locally finitary compact spaces.<\/p>\n\n\n\n

Proof.<\/em> Let us call a good<\/em> space any space X<\/em> with the above property; namely, such that for all topological spaces Y<\/em> and Z<\/em>, every separately continuous function f<\/em> : X<\/em> \u00d7 Y<\/em> \u2192 Z<\/em> is jointly continuous. The previous proposition states that every locally finitary compact space is good.<\/p>\n\n\n\n

Conversely, let X<\/em> be any good space. For Z<\/em>, we take Sierpi\u0144ski space S<\/strong>, the dcpo {0 < 1} with its Scott topology. For Y<\/em>, we consider O<\/strong>p<\/sub>X<\/em>, namely the set of all open subsets of X<\/em>, with the pointwise<\/em> topology (not the Scott topology). The latter is defined as being generated by the subbasic open sets [x<\/em> \u2208] \u225d {U<\/em> \u2208 O<\/strong>X<\/em> | x<\/em> \u2208 U<\/em>}, where x<\/em> ranges over the points of X<\/em>. We have already seen that space when we studied the double powerspace<\/a> constructions.<\/p>\n\n\n\n

Finally, we let f<\/em> : X<\/em> \u00d7 Y<\/em> \u2192 Z<\/em> be the characteristic function of the membership relation \u2208. In other words, f<\/em>(x<\/em>, U<\/em>) is equal to 1 if x<\/em> \u2208 U<\/em>, to 0 otherwise. We verify that this is a separately continuous function. There is only one non-trivial open set in Z<\/em>, which is {1}, hence it suffices to check the inverse images of {1} under f<\/em>(x<\/em>,_) and under f<\/em>(_,U<\/em>).<\/p>\n\n\n\n