Definition 10.1.
A subset \(C\) of a metric space \(X\) is closed if its complement \(X \setminus C\) is open.
Section Introduction
Once we have defined open sets in metric spaces, it is natural to ask if there are closed sets. Recall that closed intervals are important in calculus because every continuous function on a closed interval attains an absolute maximum and absolute minimum value on that interval. If we have closed sets in metric spaces, we might consider if there is some result that is similar to this for continuous functions on closed sets. In this section we introduce the idea of closed sets in metric spaces and discover a few of their properties.
Every interval of the form \([a,b]\) in \(\R\) is a closed set using the Euclidean metric. What distinguishes these closed intervals from the open intervals is that the open intervals do not contain either of their endpoints — this is what makes an open interval a neighborhood of each of its points. In general, what makes open sets open is that they do not contain their boundaries. If an open set doesn’t contain its boundary, then its complement, by contrast, should contain its boundary. This leads to the definition of a closed set.
We said that open sets are open because they do not contain their boundary and closed sets are closed because they do contain their boundary. However, we did not define what we mean by boundary. The point \(a\) on the “boundary” of an open interval of the form \(O=(a,b)\) in \(\R\) with the Euclidean metric has the property that every open ball that contains \(a\) contains points in \(O\) and points not in \(O\text{.}\) This is what makes the point \(a\) lie on the boundary. We can also think of the point \(a\) as being at the very limit of the set \(O\text{.}\) This motivates the next definition.
Definition 10.2.
Let \(X\) be a metric space, and let \(A\) be a subset of \(X\text{.}\) A boundary point of \(A\) is a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) and a point in \(X \setminus A\text{.}\)
For example, in \(A=(0,1)\) as a subset of \((\R, d_E)\text{,}\) the number 0 is a boundary point of \(A\) because any open interval in \(\R\) that contains \(0\) contains points in \(A\) and points not in \(A\text{.}\) Boundary points can arise in other ways. If \(A = \{0,1\}\) as a subset of \((\R, d_E)\text{,}\) then 0 is again a boundary point because any open interval in \(\R\) that contains \(0\) contains a point (\(0\)) in \(A\) and points not in \(A\text{.}\) However, \(0\) is the only point in \(A\) that is contained in any open interval that contains \(0\text{.}\) In this case we call \(0\) an isolated point of \(A\text{,}\) and in the case of the set \(A = (0,1)\) we call \(0\) an accumulation point or a limit point of \(A\) (the use of the word “limit” here will become clear later).
Definition 10.3.
Let \(X\) be a metric space, and let \(A\) be a subset of \(X\text{.}\)
- An accumulation point or limit point of \(A\) is a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) different from \(x\text{.}\)
- An isolated point of \(A\) is a point \(a \in A\) such that there exists a neighborhood \(N\) of \(a\) in \(X\) with \(N \cap A = \{a\}\text{.}\)
You might wonder about the use of the term “limit point” and how limit points might be related to limits. As we will see later, limit points are limits of sequences, but the definition as we have given is one that will translate directly to topological spaces later.
Note that every boundary point is either an accumulation point or an isolated point. The proof is left as an exercise.
Preview Activity 10.1.
(a)
For each of the given sets \(A\text{,}\) find all boundary points, limit points, and isolated points. Then determine if the set \(A\) is a closed set in the metric space \((X,d)\text{.}\) Explain your reasoning.
(i)
\(X = \R\text{,}\) \(d = d_E\text{,}\) the Euclidean metric, \(A = [0,0.5)\text{.}\)
(ii)
\(X = \{x \in \R \mid 0 \lt x \leq 1\}\text{,}\) \(d = d_E\text{,}\) the Euclidean metric, \(A = (0,0.5]\text{.}\)
(iii)
\(X = \{a,b,c,e\}\text{,}\) \(d\) is the discrete metric defined by
\begin{equation*}
d(x,y) = \begin{cases}0 \amp \text{ if } x = y \\ 1 \amp \text{ if } x \neq y, \end{cases}
\end{equation*}
and \(A = \{a,b\}\text{.}\)
(b)
Label each of the following statements as either true or false. If true, provide a convincing argument. If false, provide a specific counterexample.
(i)
Every limit point is a boundary point.
(ii)
Every boundary point is a limit point.
(iii)
Every limit point is an isolated point
(iv)
Every isolated point is a limit point.
(v)
Every boundary point is an isolated point.
(vi)
Every isolated point is a boundary point.
