TITLE: Long line fundamental groupoid
QUESTION [11 upvotes]: This question got me thinking about what makes the fundamental group (or groupoid) tick.  What is so special about the circle?  As another possible candidate for generalization, what about taking the one point compactification of the long closed ray R and thinking about homotopy theory with R in place of the interval?  Would a theory of "long homotopy" arise?  As a follow up, if this doesn't work, are there any other interesting instances of replacing the unit interval with another topological space and getting an interesting homotopy theory out of it?  If not is there some characterization of the interval as the unique space which induces a nice homotopy theory?

REPLY [2 votes]: Perhaps the article of J. Cannon and G. Conner, "The big fundamental group, big Hawaiian earrings, and the big free groups", will interest you. I believe they work with just what you said, the one-point compactification of the long closed ray, or something very similar.