In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales:

...the single most important fact which distinguishes locales from spaces: the fact that every locale has a smallest dense sublocale. If you want to 'sell' locale theory to a classical topologist, it's a good idea to begin asking him to imagine a world in which any intersection of dense subspaces would always be dense. Once he has contemplated some of the wonderful consequences that would follow from this result you can tell him this world is exactly the category of locales.

Q. My classical topology in somewhat rusty and neither Johnstone nor the book in which this quote is embedded in expand upon 'the wonderful consequences'. What might they be?

It seems to me that one obvious result that would be a triviality is Baires Category theorem. Also, given that position and momentum observables are represented by densely defined unbounded operators, is this result useful there, directly or indirectly?