internal point


Let X be a vector space and SX. Then xS is called an internal point of S if and only if the intersectionMathworldPlanetmath of each line in X through x and S contains a small interval around x.

That is x is an internal point of S if whenever yX there exists an ϵ>0 such that x+tyS for all t<ϵ.

If X is a topological vector spaceMathworldPlanetmath and x is in the interior of S, then it is an internal point, but the converseMathworldPlanetmath is not true in general. However if Sn is a convex set then all internal points are interior points and vice versa.


  • 1 H. L. Royden. . Prentice-Hall, Englewood Cliffs, New Jersey, 1988
Title internal point
Canonical name InternalPoint
Date of creation 2013-03-22 14:25:04
Last modified on 2013-03-22 14:25:04
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 5
Author jirka (4157)
Entry type Definition
Classification msc 52A99