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In Johnstone´s Sketches of an Elephant Volume 2, page 716,

lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the locale product $X\times Y$ is spatial.

Is this lemma still valid for a family of locally compact spatial locales, that is, if $\{X_{\alpha}\}_{\alpha}$ is a family of locally compact spatial locales is the locale $\prod_{\alpha}X_{\alpha}$ spatial?

Or anyone knows a modification for a countable family of locally compact spatial locales?

Thanks in advanced.

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    $\begingroup$ Please change the title to something more descriptive, e.g., "Spatiality of products of locally compact locales" or something similar. $\endgroup$
    – Todd Trimble
    Commented Sep 6, 2018 at 15:49
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    $\begingroup$ Also could you please formulate the statement that you have in mind? Do you mean localic coproduct or product or what? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Sep 6, 2018 at 16:01
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    $\begingroup$ I mean what do you intend to do with that family? Take its coproduct and multiply by $Y$? Or take its product and multiply by $Y$? Or multiply each of the $X_\alpha$ by $Y$ and then take coproduct of these products? Or multiply and then take product? You ask about validity of a statement but I cannot figure out what this statement is, sorry. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Sep 6, 2018 at 17:27
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    $\begingroup$ It seem rather clear to me that the question is about wether an infinite product of spatial and locally compact locales is spatial. Product mean product in the category of locales (hence coproduct in the category of frame). $\endgroup$
    – Simon Henry
    Commented Sep 6, 2018 at 19:42
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    $\begingroup$ @AngelZaldívar : the results about countably presented locales being spatial can be found in S.Valentini "Every countably presented formal topology is spatial, classically " projecteuclid.org/euclid.jsl/1146620155 A product of countably generated locale is again countably generated and it should be possible to show that separable metric spaces are countably generated locales, though I don't know a references for that. $\endgroup$
    – Simon Henry
    Commented Sep 7, 2018 at 11:40

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