Inner point

x is the inner point of S, y is the edge point.

Inner point and inner or open core are terms from topology , a branch of mathematics .

Each element of a subset of a topological space , a to the environment in can find the fully located, is an interior point of . The set of all interior points of is called interior or open core of . ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$

Example: If you consider a circular disk as part of the plane, then the points on the edge of the circle are not inner points (but edge points ). In contrast, all points between the edge of the circle and the center of the circle and the center of the circle are inner points of the area of ​​the circle.

definition

Let be any subset of a topological space . Then a point from is an interior point of if and only if there is a neighborhood of in , i.e. H. if there is a subset that contains and is open in . ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle x_{0}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle x_{0}}$ ${\displaystyle X}$ ${\displaystyle U\subseteq M}$ ${\displaystyle x_{0}}$ ${\displaystyle X}$

The set of all interior points of is called interior or open core of ; it is the largest open subset of . It is usually referred to with or, especially in English-language literature, with or . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M^{\circ }}$ ${\displaystyle \operatorname {Int} (M)}$ ${\displaystyle \operatorname {int} (M)}$

properties

• A subset of a topological space is open exactly when it is equal to its interior.
• The inside of the complement is the complement of the conclusion and vice versa:

${\displaystyle (X\setminus M)^{\circ }=X\setminus {\overline {M}}}$ and ${\displaystyle {\overline {X\setminus M}}=X\setminus M^{\circ }}$

The interior of the complement is also the exterior of M . The space X thus falls into affairs, border and exterior of M .

example

Take the following amount and number : ${\displaystyle M}$ ${\displaystyle a}$

${\displaystyle a}$is an inner point of because there is one such that is a subset of : ${\displaystyle M}$ ${\displaystyle \epsilon >0}$ ${\displaystyle ]a-\epsilon ,a+\epsilon [}$ ${\displaystyle M}$

Inner points of intervals

• The inner points of the compact interval are exactly the points belonging to the open interval . ${\displaystyle \left[a,b\right]}$ ${\displaystyle \left(a,b\right)}$
• Likewise, the inner points of the half-open interval or the half-open interval are exactly the points belonging to the open interval . ${\displaystyle \left[a,b\right)}$ ${\displaystyle \left(a,b\right]}$ ${\displaystyle \left(a,b\right)}$
• All points of the open interval are inner points. ${\displaystyle \left(a,b\right)}$

See also

• Core operator
• In particular in the context of convex subsets of real vector spaces one considers the interior with respect to the affine hull ; one then speaks of relatively internal points .

literature

• Boto von Querenburg : Set theoretical topology. 3rd revised and expanded edition. Springer-Verlag, Berlin et al. 2001, ISBN 3-540-67790-9 ( Springer textbook ).