So, I don't quite know what as you wrote it is, but if you let then by definition (being just an interval) one has that is open in with the dictionary ordering, and since we have that our set in question is the intersection of an open set in with and so, by definition, open in the subspace topology. I also am not quite sure what your last paragraph means, please try to reexplain it--I believe it's what I'm about to say though. To me what makes the most obvious sense is that if we suppose that to be our set in question, then if were open in with the dictionary ordering we can find some basic open set with . But, since is not a maximal member of we know that we can take to be an open interval, and so in particular there exists with which contradicts . More visually appealing, any open set containing spills onto the next vertical line in .