Section Introduction
Open sets are vitally important in topology. We will see later that every topological space is completely defined by its open sets, and continuous functions can be defined just in terms of open sets. In this section we introduce the idea of open balls and neighborhoods in metric spaces and discover a few of their properties. This discussion will form the basis for introducing open sets in the next section.
Recall that the continuity of a function \(f\) from a metric space \((X, d_X)\) to a metric space \((Y, d_Y)\) at a point \(a\) is defined in terms of sets of points \(x \in X\) such that \(d_X(x,a) \lt \delta\) and \(y \in Y\) such that \(d_Y(y, f(a)) \lt \epsilon\) for positive real numbers \(\delta\) and \(\epsilon\text{.}\) In \(\R\) with the Euclidean metric \(d_E\text{,}\) for real numbers \(x\) and \(a\) the set of \(x\) values satisfying \(d_E(x,a) \lt \delta\) is the set of \(x\) values so that \(| x-a | \lt \delta\text{.}\) We often write this set in interval notation as \((a-\delta,
a+\delta)\) and call \((a-\delta,
a+\delta)\) an open interval. An informal reason that we call such an interval open (as opposed to the intervals \([a-\delta,
a+\delta)\text{,}\) \((a-\delta, a+\delta]\text{,}\) or \([a-\delta,
a+\delta]\)) is that the open interval does not contain either of its endpoints. A more substantial reason to call such an interval open is that if \(x'\) is any element in \((a-\delta,
a+\delta)\text{,}\) then we can find another open interval around \(x'\) that is completely contained in the interval \((a-\delta,
a+\delta)\text{.}\) So you could naively think of an open interval as one in which there is enough room in the interval for any point in the interval to wiggle around a bit and stay within the interval.
Since the open interval \((a-\delta,
a+\delta)\) can be described completely by the Euclidean metric as the set of \(x\) values so that \(d_E(x,a) \lt \delta\text{,}\) there is no reason why we can’t extend this notation of open interval to any metric space. We must note, though, that \(\R\) is one-dimensional while most metric spaces are not, so the term “interval” will no longer be appropriate. We replace the concept of interval with that of an open ball.
Definition 7.1.
Let \((X, d_X)\) be a metric space, and let \(a \in X\text{.}\) For \(\delta \gt 0\text{,}\) the open ball \(B(a, \delta)\) of radius \(\delta\) around \(a\) is the set
\begin{equation*}
B(a, \delta) = \{x \in X \mid d_X(x,a) \lt \delta\}\text{.}
\end{equation*}
We note here that our notation for an open ball is not universal. For example, some texts use \(B_{\delta}(a)\) for our \(B(a,\delta)\text{.}\)
Preview Activity 7.1.
Describe and draw a picture of the indicated open ball in each of the following metric spaces.
(a)
The open ball \(B(2, 1)\) in the metric space \((\R, d_E)\) with the Euclidean metric
\begin{equation*}
d_E(x,y) = | x-y |\text{.}
\end{equation*}
(b)
The open ball \(B((3,2), 1)\) in the metric space \((\R^2, d_E)\) with the Euclidean metric
\begin{equation*}
d_E((x_1,x_2),(y_1,y_2)) = \sqrt{(x_1-y_1)^2 + (x_2-y_2)^2}\text{.}
\end{equation*}
(c)
The open ball \(B((3,2), 1)\) in the metric space \((\R^2, d_M)\) with the max metric
\begin{equation*}
d_M((x_1,x_2),(y_1,y_2)) = \max\{| x_1-y_1 |, | x_2-y_2 |\}\text{.}
\end{equation*}
(d)
The open ball \(B((3,2), 1)\) in the metric space \((\R^2, d_T)\) with the taxicab metric
\begin{equation*}
d_T((x_1,x_2),(y_1,y_2)) = | x_1-y_1 | + | x_2-y_2 |\text{.}
\end{equation*}
(e)
The open ball \(B((3,2), 1)\) in the metric space \((\R^2, d)\) with the discrete metric
\begin{equation*}
d(x,y) = \begin{cases}0 \amp \text{ if } x=y \\ 1 \amp \text{ if } x \neq y. \end{cases}\text{.}
\end{equation*}
What is the difference between \(B((3,2),1)\) and \(B((3,2),r)\) in this metric space if \(r > 1\text{?}\) If \(r \lt 1\text{?}\)
