# Spatiality of products of locally compact locales

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?

• 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. – მამუკა ჯიბლაძე Sep 6 '18 at 17:27