Section Limit Points, Boundary Points, Isolated Points, and Sequences
Recall that a limit point of a subset \(A\) of a metric space \(X\) is a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) different from \(x\text{.}\) You might wonder about the use of the word “limit” in the definition of limit point. The next activity should make this clear.
Activity 10.6.
Let \(X\) be a metric space, let \(A\) be a subset of \(X\text{,}\) and let \(x\) be a limit point of \(A\text{.}\)
(a)
Let \(n \in \Z^+\text{.}\) Explain why \(B\left(x, \frac{1}{n}\right)\) must contain a point \(a_n\) in \(A\) different from \(x\text{.}\)
(b)
What is \(\lim a_n\text{?}\) Why?
The result of
Activity 10.6 is summarized in the following theorem.
Theorem 10.6.
Let \(X\) be a metric space, let \(A\) be a subset of \(X\text{,}\) and let \(x\) be a limit point of \(A\text{.}\) Then there is a sequence \((a_n)\) in \(A\) that converges to \(x\text{.}\)
Of course, the constant sequence \((a)\) always converges to the point \(a\text{,}\) so every point in a set \(A\) is the limit of a sequence. With limit points there is a non-constant sequence that converges to the point. We might ask what we can say about a point \(a \in A\) if the only sequences in \(A\) that converges to \(a \in A\) are the eventually constant sequences \((a)\text{.}\) (By an eventually constant sequence \((a_n)\text{,}\) we mean that there is a positive integer \(K\) such that for \(k \geq K\text{,}\) we have \(a_k = a\) for some element \(a\text{.}\)) That is the subject of our next activity.
Activity 10.7.
Let \((X,d)\) be a metric space, and let \(A\) be a subset of \(X\text{.}\)
(a)
Let \(a\) be an isolated point of \(A\text{.}\) Prove that the only sequences in \(A\) that converge to \(a\) are the eventually constant sequences \((a)\text{.}\)
(b)
Prove that if the only sequences in \(A\) that converges to \(a\) are the eventually constant sequences \((a)\text{,}\) then \(a\) is an isolated point of \(A\text{.}\)
Boundary points are points that are, in some sense, situated “between” a set and its complement. We will make this idea of “between” more concrete soon.
An argument just like the one in
Activity 10.6 gives us the following result about boundary points.
Theorem 10.7.
Let \(X\) be a metric space, let \(A\) be a subset of \(X\text{,}\) and let \(b\) be a boundary point of \(A\text{.}\) Then there are sequences \((x_n)\) in \(X \setminus A\) and \((a_n)\) in \(A\) that converge to \(x\text{.}\)
