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Section Summary
Important ideas that we discussed in this section include the following.
A subset \(A\) of a metric space \((X,d)\) is a metric space, called a subspace, by using the metric \(d|_{A \times A}\) on \(A\text{.}\)
If \(X\) is a metric space and \(A\) is a subspace of \(X\text{,}\) a subset \(O_A\) of \(A\) is open in \(A\) if and only if \(O_A = X \cap O\) for some open set \(O\) in \(X\text{.}\) A subset \(C_A\) of \(A\) is closed in \(A\) if \(C_A = A \cap O_A\) for open set \(O_A\) in \(A\text{.}\) Alternatively, a set \(C_A\) is closed in \(A\) if \(C_A = A \cap C\) for some closed set \(C\) in \(X\text{.}\)
Let \((X_i, d_i)\) be metric spaces for \(i\) from \(1\) to some positive integer \(n\text{.}\) The product metric space \((X,d)\) is the Cartesian product
\begin{equation*}
X = X_1 \times X_2 \times \cdots \times X_n = \prod_{i=1}^n X_i
\end{equation*}
with metric \(d\) defined by
\begin{equation*}
d(x,y) = \sqrt{\sum_{i=1}^n d_i(x_i,y_i)^2}
\end{equation*}
when \(x = (x_1, x_2, \ldots,
x_n)\) and \(y = (y_1, y_2, \ldots, y_n)\) are in \(X\text{.}\)
