# 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 InternalPoint 2013-03-22 14:25:04 2013-03-22 14:25:04 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 52A99