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

.