Accumulation Points aka Limit Points, Real Analysis II
In the last video, we defined closed sets as those whose complement is open. We explored examples to get a feel for what closed sets look like. In this lesson, we introduce the concept of accumulation points, which gives us another way to determine if a set is closed.
(MA 426 Real Analysis II, Lecture 9)
An accumulation point of a set A in a metric space is a point x where every open neighborhood around x contains at least one point from A that is not x itself. Importantly, x does not need to belong to A. Accumulation points are also known as limit points or cluster points.
To illustrate this, we worked through several examples. For the interval [0,1), we found that the set of accumulation points is [0,1]. For the set of rational numbers, every real number is an accumulation point. The set of integers, however, has no accumulation points because we can always find an open neighborhood around any integer that contains no other integers. The Archimedean set, consisting of points like 1/n, has 0 as its only accumulation point.
We also explored how the shape of neighborhoods changes in different metric spaces, such as the xy-plane or a discrete metric space. For example, the graph of y = sin(1/x) for positive x has accumulation points along the y-axis between -1 and 1, as well as all points on the graph itself.
Finally, we discussed a key theorem: a set A is closed if and only if it contains all its accumulation points. This means that if any accumulation point of A belongs to its complement, A is not closed. This theorem links our earlier examples to the concept of closed sets and helps us determine if certain sets are or aren’t closed.
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