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In a metric space an arc which
is isometric to a real interval is called a segment. In this paper it is shown
that, for 1 ≦ n ≦ 3, n-dimensional Euclidean space (En) is topologically
characterized, among locally compact, n-dimensional spaces, by admitting
a metric with the following properties: (1) every two points of the space
are endpoints of a unique segment, (2) if two segments have an endpoint
and one other point in common then one is contained in the other and (3)
every segment can be extended, at either end, to a larger segment. This
follows from the more general result that, for 1 ≦ n ≦ 3, a locally compact,
n-dimensional space which admits a metric with properties (1) and (2) is
homeomorphic to an n-manifold lying between the closed n-ball and its
interior.
Property (1) suffices to characterize En, for n = 1 or 2, among locally compact,
locally homogeneous, n-dimensional spaces. For n > 3, properties (1), (2), and (3)
characterize En among locally compact, n-dimensional spaces that contain a
homeomorph of an n-ball.
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