internal point

Definition.

Let $X$ be a vector space and $S\\subset X$ . Then $x\\in S$ is called aninternal 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

单词	InternalPoint
释义	internal point Definition. Let $X$ be a vector space and $S\\subset X$ . Then $x\\in S$ is called aninternal 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
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