I answered this question on "is there a longest geodesic" by a kind of a joke, which I couldn't resist: the long line! Simply going by the name it had to be the 'longest geodesic'! I didn't bother explaining the joke, after all, a joke that requires explaining, is no joke at all, in my exoerience.
I cinsidered it a joke because I didn't know of a theory of such 'large' manifolds and I thought my intentions would be clear - they weren't apparently.
But then, much to my surprise, whilst researching something else entirely, I came upon a classification of such 'large' surfaces! Unfortunately I didn't make a note of it. However, whilst trying to find where and what this result exactly is, I came upon a paper by Rafael Dahmen on embeddings of the long line in weakly complete vector spaces. These are spaces like $R^I$ for arbitrary $$. The paper is here and titled: Smooth Embeddings of the Long Line and other Non-Paracompact Manifolds into Locally Convex Spaces.
Q. What I'm after is a paper or a book outlining the classification of such large surfaces.
I recall the source that I referenced a certain book, a handbook, from what I recall. Unfortunately, I don't remember the title of the book or which source it was - other than the source was online. Hence the question.
