Boundary of a set in a metric space, Real Analysis II
In this video, we cover the definition of the boundary of a set in a metric space (M,d). The boundary of a set A, denoted as the boundary of A, is defined as the intersection of the closure of A and the closure of the complement of A. This definition shows that the boundary of a set is closed and that the boundary of A is the same as the boundary of its complement, A^c.
(MA 426 Real Analysis II, Lecture 11)
Then we work through several examples to compute the boundary of different sets, such as the entire metric space, the set of integers, the set of rational numbers, intervals on the real number line, the Archimedean set, quadrants in the xy-plane, and subsets of a discrete metric space.
We also discuss and prove two alternative characterizations of the boundary of a set. The first states that a point x is in the boundary of A if every open neighborhood of x intersects both A and its complement. The second alternative characterizes the boundary of a set as the closure of the set minus its interior.
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