Topology: An Inquiry-Based Approach

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.

The result of Activity 10.6 is summarized in the following theorem.

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 $k\ge K\text{,}$ we have $a_k=a$ for some element $a\text{.}$) That is the subject of our next activity.

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.