Interior of a set in a metric space, Real Analysis II
In this lesson, we explore more about open sets in metric spaces. You should already be familiar with key concepts like metric spaces, open sets, neighborhoods, epsilon neighborhoods, and the basic properties of open sets. We begin by introducing the concepts of interior points and the interior of a set. An interior point of a set A is a point where there exists an open set around it that is fully contained within A. The collection of all interior points forms what we call the interior of A.
(MA 426 Real Analysis II, Lecture 7)
We then work through several examples to solidify these definitions. We first consider the interior of R^n, where we find that the interior is the entire space R^n itself. We extend this analysis to subsets of the real number line and a discrete metric space, illustrating how the concept of interior works differently in various settings.
Following these examples, we prove some key results. We prove that the interior of a set A is the union of all open subsets of A, and therefore, the interior is always open. We also establish that a set A is open if and only if it is equal to its own interior.
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