internal point

Definition.

Let $X$ be a vector space and $S\subset X$ . Then $x\in S$ is called an internal point of $S$ if and only if the intersection 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 $y\in X$ there exists an $\epsilon>0$ such that $x+ty\in S$ for all $t<\epsilon$ .

If $X$ is a topological vector space and $x$ is in the interior of $S$ , then it is an internal point, but the converse is not true in general. However if $S\subset{\mathbb{R}}^{n}$ is a convex set then all internal points are interior points and vice versa.

References

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