No CrossRef data available.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 24 October 2008
§ 1. “Stretches” joining Points in a Metrical Space.
If x, y, z are any points in a metrical space, then by definition
Regarding x and y as fixed, we may define the set of points z for which
as the “stretch joining the points xy” or more shortly the stretch xy*. Then for points z not on the stretch,
The stretch xy always contains the points x and y themselves, which we may call the “end-points” of the stretch. In particular cases, it may contain no other points: e.g. with the function δ0(x, y) = 1 when xǂy, each stretch contains only its end-points †.
* See Menger, , “Ueber geodätische Linien in allgemeinen metrischen Ränmen,” Amsterdam Proceedings, Vol. xxix (1926), p. 166.Google Scholar
† If each stretch contains at least one" point in addition to its end-points, the space is said to be “convex.” See Menger, op. cit.