Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-17T02:47:47.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Lines and Planes in a Metrical Space

Published online by Cambridge University Press:  24 October 2008

W. Hunter
W. Hunter
Affiliation:
Trinity College
Get access Rights & Permissions [Opens in a new window]

Extract

§ 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 †.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 6 , April 1927 , pp. 653 - 674
Copyright
Copyright © Cambridge Philosophical Society 1927

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

* 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.