Theorem 11.2.
Let \((X,d)\) be a metric space and \(A\) a subset of \(X\text{.}\) A subset \(O_A\) of \(A\) is open in \(A\) if and only if there is an open set \(O_X\) in \(X\) so that \(O_A = O_X \cap A\text{.}\)
Section Open and Closed Sets in Subspaces
We saw in our preview activity that a subspace does not necessarily inherit the properties of the larger space. For example, a subset of a subspace might be open in the subspace, but not open in the larger space. However, there is a connection between the open subsets in a subspace and the open sets in the larger space.
Proof.
Let \((X,d)\) be a metric space and \(A\) a subset of \(X\text{.}\) Let \(O_A\) be an open subset of \(A\text{.}\) So for each \(a \in A\) there is a \(\delta_a \gt 0\) so that \(B_A(a, \delta_a) \subseteq O_A\text{,}\) where \(B_A(a, \delta_a)\) is the open ball in \(A\) centered at \(a\) of radius \(\delta_a\text{.}\) Then, \(O_A~=~\bigcup_{a \in O_A}~B_A(a, \delta_a)\text{.}\) Now let \(B_X(a, \delta_a)\) be the open ball in \(X\) centered at \(a\) of radius \(\delta_a\text{,}\) and let \(O_X = \bigcup_{a \in O_A} B_X(a, \delta_a)\text{.}\) Note that
\begin{equation*}
B_A(a, \delta_a)~=~B_X(a, \delta_a)~\cap~A\text{.}
\end{equation*}
As a union of open balls in \(X\text{,}\) the set \(O_X\) is open in \(X\text{.}\) Now
\begin{align*}
O_X \cap A = \left(\bigcup_{a \in O_A} B_X(a, \delta_a) \right) \cap A \amp = \bigcup_{a \in O_A} \left( B_X(a, \delta_a) \cap A \right)\\
\amp = \bigcup_{a \in O_A} B_A(a, \delta_a) = O_A\text{.}
\end{align*}
So there is an open set \(O_X\) in \(X\) such that \(O_A = O_X \cap A\text{.}\)
For the reverse implication, see the following activity.
Activity 11.2.
Let \((X,d)\) be a metric space and \(A\) a subset of \(X\text{.}\) Suppose that \(O_A = A \cap O_X\) for some set \(O_X\) that is open in \(X\text{.}\) In this activity we will prove that \(O_A\) is an open subset of \(A\text{.}\)
(a)
Let \(a \in O_A\text{.}\) Explain why there must exist a \(\delta \gt 0\) such that \(B_X(a, \delta)\text{,}\) the open ball in \(X\) of radius \(\delta\) around \(a\) in \(X\text{,}\) is a subset of \(O_X\text{.}\)
(b)
What would be a natural candidate for an open ball in \(A\) centered at \(a\) that is contained in \(A\text{?}\) Prove your conjecture.
(c)
What conclusion can we draw?
We might now wonder about closed sets in a subspace. If \(X\) is a metric space and \(A\) is a subspace, then by definition a subset \(C_A\) of \(A\) is closed if and only if \(C_A = A \setminus O_A\) for some set \(O_A\) that is open in \(A\text{.}\) The analogy of Theorem 11.2 is true for closed sets in subspaces.
Theorem 11.3.
Let \((X,d)\) be a metric space and \(A\) a subset of \(X\text{.}\) A subset \(C_A\) of \(A\) is closed in \(A\) if and only if there is a closed set \(C_X\) in \(X\) so that \(C_A = C_X \cap A\text{.}\)
The proof is left to Exercise 4.
