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Section Summary
Important ideas that we discussed in this section include the following.
A subset \(O\) of a metric space \((X,d)\) is an open set if \(O\) is a neighborhood of each of its points. Alternatively, \(O\) is open if \(O\) is a union of open balls.
A point \(a\) in a subset \(A\) of a metric space \((X,d)\) is an interior point of \(A\) if \(A\) is a neighborhood of \(a\text{.}\) A set \(O\) is open if every point of \(O\) is an interior point of \(O\text{.}\)
The interior of a set is the set of all interior points of the set. The interior of a set \(A\) in a metric space \(X\) is the largest open subset of \(X\) contained in \(A\text{.}\) A set is open if and only if the set is equal to its interior.
A function \(f\) from a metric space \(X\) to a metric space \(Y\) is continuous if \(f^{-1}(O)\) is open in \(X\) whenever \(O\) is open in \(Y\text{.}\)
Any union of open sets is open, while any finite intersection of open sets is open.
