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Wikipedia lists eleven intervals, which I've made pages for. Here they are, along with their pages and equivalent one-dimensional manifolds. I haven't distinguished left-open and right-open because that'd just be weird.

empty: [b,a] Null polytope (Null polytope)
degenerate: [a,a] Point (Point)
proper and bounded:
open: (a,b) Open interval (Euclidean line)
closed: [a,b] Line segment (Line segment)
left-closed, right-open: [a,b) Half-open line segment (Half-open line segment)
left-open, right-closed: (a,b] Half-open line segment (Half-open line segment)
left-bounded and right-unbounded:
left-open: (a,+\infty) Open ray (Half-open line segment)
left-closed: [a,+\infty) Ray (Euclidean line)
left-unbounded and right-bounded:
right-open: (-\infty,b): Open ray (Half-open line segment)
right-closed: (-\infty,b]: Ray (Euclidean line)
unbounded at both ends (simultaneously open and closed): (-\infty,+\infty): Euclidean line (Euclidean line)

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