inner point

A circle is drawn with an inner point at its center.

Definition

1. Noun (Mathematics):
   • A point inside a region: In geometry or topology, an "inner point" refers to a point that lies within the interior of a set or region, not on its boundary. For a given set, a point is an "inner point" if there exists some small neighbourhood (e.g., a circle or ball) around it that is entirely contained within the set.

Usage Examples

• (A point located strictly inside the circle, not on its edge.)
• (A point with a surrounding area fully inside the set.)
• (Every point has a neighbourhood contained within the set.)

Advanced Usage

• "inner point of a set": In advanced mathematics, this phrase is used to describe points that are not limit points of the complement.

  • The point (0,0) is an inner point of the open disc of radius 1. (The disc's interior includes all points with distance less than 1 from the origin.)

• "inner point of an interval": In real analysis, points strictly between the endpoints of an interval.

  • For the interval (2,5), the number 3 is an inner point, but 2 is not. (2 is a boundary point, not an inner point.)

Variants and Related Words

• Interior point (n): a synonym for "inner point", more commonly used in mathematics.
  • The interior of a set consists of all its interior points. (All points that are inner points.)
• Interior (n): the set of all inner points of a given set.
  • The interior of a closed ball is an open ball. (The collection of all points strictly inside.)

Synonyms

• Interior point: a point belonging to the interior of a set.
• Internal point: a point inside a region or object (less technical).

Related Idioms