Does three dimensional Euclidean space contain unbounded closed curves that do not cross themselves?
It does not seem possible to find examples of such anmd yet it is not clear just what is standing in
the way. More precisely, let n be any positive integer not less than 3 and let E(n) be n-dimensional
Euclidean space. Does there exist a subset S of E(n) with the following properties that is not compact?
(1)S is closed, connected and locally connected. (2) If any point of S is removed, the resulting space
is still connected and is homeomorphic to a straight line. (3) If any distinct pair of points of S are
removed, the resulting space is no longer connected and has two components. Finally can S be a
non-compact subset of a separable and infinite dimensional Hilbert space?



A question about closed curves
Note that $S$ is a connected 1-dimensional manifold.
Since it is not compact we get that $S$ is homeomorphic to $\mathbb R$, 
a contradiction.


