# 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 $\u03f5>0$ such that $x+ty\in S$ for all $$.

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 |