Vol. 48, No. 1, 1973

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Metric characterizations of Euclidean spaces

Gordon Owen Berg

Vol. 48 (1973), No. 1, 11–28

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.

Mathematical Subject Classification 2000
Primary: 54F05
Secondary: 54E35
Received: 7 April 1972
Revised: 28 March 1973
Published: 1 September 1973
Gordon Owen Berg