The go-to example of elliptic space is a sphere where geodesics turn into great circles of finite length. But is it possible to have an elliptic space which doesn't 'merge' with itself once it's made a full turn? ie. infinite, unbounded, simply-connected, but with constant positive curvature everywhere.
I can't find anything like it online, but then maybe I simply don't know what word to search for?
Edit: due to user427327's remarks I thought I'd elaborate with a 1D example, nevermind that curves don't have intrinsic curvature.
The above image shows two 'spaces', both of constant positive curvature. In the left space if you travel 2pi you end up back where you started. In the right space you end up somewhere else entirely. You can keep on travelling and keep on getting further and further away from where you started, despite travelling inside a 'space' of positive curvature. Is the same not possible for 2D spaces with constant positive curvature?