Preview Activity 11.1.
Let \((X,d)\) be a metric space, and let \(A\) be a subset of \(X\text{.}\) To make the subset \(A\) into a metric space, we need to define a metric on \(A\text{.}\) For us to consider \(A\) as a subspace of \(X\text{,}\) we want the metric on \(A\) to agree with the metric on \(X\text{.}\) So we define \(d' : A \times A \to \R\) by
\begin{equation*}
d'(a_1,a_2) = d(a_1,a_2)
\end{equation*}
for all \(a_1, a_2 \in A\text{.}\) Note that \(d\) and \(d'\) are different functions because they have different domains.
(a)
Show that \(d'\) is a metric on \(A\text{.}\) Since \(d'\) is a metric on \(A\) it follows that \((A,d')\) is a metric space. The metric \(d'\) is the restriction of \(d\) to \(A \times A\) and can also be denoted by \(d_A\text{.}\)
Definition 11.1.
Let \((X,d)\) be a metric space. A subspace of \((X,d)\) is a subset \(A\) of \(X\) together with the metric \(d_A\) from \(A \times A\) to \(\R\) defined by
\begin{equation*}
d_A(a_1,a_2) = d(a_1,a_2)
\end{equation*}
for all \(a_1,a_2 \in A\text{.}\)
We might wonder what, if any, properties of the space \(X\) are inherited by a subspace.
(b)
Let \((X,d) = (\R,d_E)\) and let \(A = [0,1]\text{.}\) Let \(0 \lt a \lt 1\text{.}\) Is the set \([0,a)\) open in \(X\text{?}\) Is the set \([0,a)\) open in \(A\text{?}\) Explain.
(c)
Let \((X,d) = (\R,d_E)\) and let \(A = \Z\text{.}\) What are the open subsets of \(A\text{?}\) Explain.
(d)
Let \((X,d) = (\R^2,d_E)\text{,}\) let \(A = \{(x,0) \mid x \in \R\}\) (the \(x\)-axis in \(\R^2\)), and let \(Z = \{(x,0) \mid 0 \lt x \lt 1\}\text{.}\) Note that \(Z \subset A \subset X\) and we can consider \(Z\) as a subspace of \(A\) and \(X\text{,}\) and \(A\) as a subspace of \(X\text{.}\)
(i)
Explain why \(A\) is a closed subset of \(X\text{.}\)
(ii)
Explain why \(Z\) is an open subset of \(A\text{.}\)
(iii)
Is \(Z\) an open subset of \(X\text{?}\) Explain
