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Section Summary
Important ideas that we discussed in this section include the following.
If \((X,d)\) is a metric space and \(a \in X\text{,}\) then an open ball centered at \(a\) is a set of the form
\begin{equation*}
B(a,\delta) = \{ x \in X \mid d(x,a) \lt \delta\}
\end{equation*}
for some positive number \(\delta\text{.}\)
A subset \(N\) of a metric space \((X,d)\) is s neighborhood of a point \(a \in N\) if there is a positive real number \(\delta\) such that \(B(a,\delta) \subseteq N\text{.}\)
An important property of open balls is that every open ball is a neighborhood of each of its points. This is our first step toward defining the concept of open sets that will form the foundation for topological spaces.
A function \(f\) from a metric space \((X,d_X)\) to a metric space \((Y,d_Y)\) is continuous at \(a \in X\) if \(f^{-1}(N)\) is a neighborhood of \(a\) in \(X\) for any neighborhood \(N\) of \(f(a)\) in \(Y\text{.}\)
