Interior, Exterior and Boundary of a set.

By | September 12, 2023

In Topological mathematics, the concept “interior,” “exterior,” and “boundary” are used to describe different parts of a set in relation to its surrounding space. In this article we will define each term and with illustrated examples to give their clear meanings.

  1. Interior:
    The interior of a set can be explained as the collection of all points that lies entirely within the set and the do not touch or lie on the boundary. It consists of the “core” or the “interior points” of the set. In other words, it includes all the points that can be surrounded by an open neighborhood entirely contained within the set. The interior of a set A is denoted as Int(A).

Example:
Consider the set K = (1, 2), which represents the open interval between 1 and 2. In this case, the interior of K is also (1, 2) because all the points within this interval do not touch the endpoints 1 and 2.

  1. Exterior:
    The exterior of a set is the set of points that lie outside of the set and are not part of its closure. It consists of all the points that can be surrounded by an open neighborhood entirely contained outside the set. Symbolically, we write the exterior of a set as Ext.

Example:
considering the set A = [0, 1], representing the closed interval including both endpoints. In this case, the exterior of A would be the collection of all points outside the interval, such as numbers less than 0 (negative numbers) or non negative numbers (numbers greater than 1).

  1. Boundary:
    The boundary of a set can be explained as the collection of points that are neither entirely in the interior nor entirely in the exterior but on the edge. It consists of the points that “lie on the edge” of the set and are shared by both the set and its complement. The boundary of a set is denoted as ∂A. Some authors use bd(A) ⁡ , fr(A) ⁡ , Ab or β(A) as boundary of a set.
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Example:
For the set U = [0, 1) ∪ (1, 2], which represents the union of the closed interval [0, 1) and the open interval (1, 2], the boundary consists of the points 0 and 1, as they are common to both the interior and exterior of the set.

In conclusion, the interior of a set consists of points inside the set, the exterior consists of points outside the set, and the boundary consists of points on the edge of the set.

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