I was perusing a problem for a project I am doing, and the set up is kind of a mouthful. But basically we have a boolean algebra $B$; for each $x$ in $B$, let $M_x$ be the set of maximal ideals of $B$ that don't contain $x$. And let $M$ be the set of all maximal ideals of $B$.
Now the book just remarks that $\{M_x \mid x\in B\}$ can be all the open sets of a topology on M. However, I don't quite see how it's closed under infinite union, since, well, $B$ isn't necessarily complete, right? Any pointers would be appreciated, as I am a little bit stumped.
(FYI, the screenshot is from Lattices and Ordered Algebraic Structures by TS Blyth.)
Indeed, the statement in the book is wrong (though it appears from the following sentence that the author had the correct statement in mind and just wrote it wrong). Except in trivial cases (when $B$ is finite), $\{M_x:x\in B\}$ will not be closed under infinite unions. (Even when $B$ is complete it will not be closed under infinite unions, since the map $x\mapsto M_x$ does not turn infinite joins into infinite unions; that only works for finite joins.) The correct topology to take is instead the topology generated by $\{M_x:x\in B\}$, so an open set is a union of sets of the form $M_x$ (since such sets are closed under finite intersections).