Definition 8.1.
A subset \(O\) of a metric space \(X\) is an open set if \(O\) is a neighborhood of each of its points.
Section Introduction
Consider the interval \((a,b)\) in \(\R\) using the Euclidean metric. If \(m = \frac{a+b}{2}\text{,}\) then \((a,b) = B\left(m,\frac{b-a}{2}\right)\text{,}\) so every open interval is an open ball. As an open ball, an open interval \((a,b)\) is a neighborhood of each of its points. This is the foundation for the definition of an open set in a metric space.
Recall that we defined a subset \(N\) of \(X\) to be neighborhood of point \(a\) in a metric space \((X,d)\) if \(N\) contains an open ball \(B(a, \epsilon)\) for some \(\epsilon \gt 0\text{.}\) We saw that every open ball is a neighborhood of each of its points, and we will now extend that idea to define an open set in a metric space.
So, by definition, any open ball is an open set. Also by definition, open sets are neighborhoods of each of their points. Open sets are different than non-open sets. For example, \((0,1)\) is an open set in \(\R\) using the Euclidean metric, but \([0,1)\) is not. The reason \([0,1)\) is not an open set is that there is no open ball centered at \(0\) that is entirely contained in \([0,1)\text{.}\) So \(0\) has a different property than the other points in \([0,1)\text{.}\) The set \([0,1)\) is a neighborhood of each of the points in \((0,1)\text{,}\) but is not a neighborhood of \(0\text{.}\) We can think of the points in \((0,1)\) as being in the interior of the set \([0,1)\text{.}\) This leads to the next definition.
Definition 8.2.
Let \(A\) be a subset of a metric space \(X\text{.}\) A point \(a \in A\) is an interior point of \(A\) if \(A\) is a neighborhood of \(a\text{.}\)
As we will soon see, open sets can be characterized in terms of interior points.
Preview Activity 8.1.
(a)
Determine if the set \(A\) is an open 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)
\(X = \{x \in \R \mid 0 \leq x \leq 1\}\text{,}\) \(d = d_E\text{,}\) the Euclidean metric, \(A = [0,0.5)\text{.}\) Assume that the Euclidean metric is a metric on \(X\text{.}\)
(iii)
\(X = \{a,b,c,d\}\text{,}\) \(d\) 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)
(i)
What are the interior points of the following sets in \((\R, d_E)\text{?}\) Explain.
\begin{equation*}
(0,1) \ \ \ (0,1] \ \ \ [0,1) \ \ \ [0,1]\text{.}
\end{equation*}
(ii)
Let \(A = \{0, 1, 2\}\) in \((\R, d_E)\text{.}\) What are the interior points of \(A\text{?}\) Explain.
(iii)
Let \(\Q\) be the set of rational numbers in \((\R, d_E)\text{.}\) What are the interior points of \(\Q\text{?}\) Explain.
