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Chapter 11: Problem 8
Show that if \(G\) is an open set and \(F\) is a closed set, then \(G \backslash F\) is an open set and \(F \backslash G\) is a closed set.
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Show that if \(d\) is the discrete metric on a set \(S\), then every subset of \(S\) is both open and closed in \((S, d)\).
Using the notation of the preceding exercise, let \(A, B\) be sets in \(\mathbb{R}\). Show that we have \(A \subseteq\) \(A^{-},\left(A^{-}\right)^{-}=A^{-}\), and that \((A \cup B)^{-}=A^{-} \cup B^{-}\). Show that \((A \cap B)^{-} \subseteq A^{-} \cap B^{-}\), and give an example to show that the inclusion may be proper.
A point \(x \in \mathbb{R}\) is said to be an interior point of \(A \subseteq \mathbb{R}\) in case there is a neighborhood \(V\) of \(x\) such that \(V \subseteq A\). Show that a set \(A \subseteq \mathbb{R}\) is open if and only if every point of \(A\) is an interior point of \(A\).
Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=x^{2}\) for \(x \in \mathbb{R}\) (a) Show that the inverse image \(f^{-1}(I)\) of an open interval \(I:=(a, b)\) is either an open interval. the union of two open intervals, or empty, depending on \(a\) and \(b\). (b) Show that if \(I\) is an open interval containing 0 , then the direct image \(f(I)\) is not open.
If \(A \subseteq \mathbb{R}\), let \(A^{-}\) be the intersection of all closed sets containing \(A\); the set \(A^{-}\) is called the closure of \(A\). Show that \(A^{-}\) is a closed set, that it is the smallest closed set containing \(A\), and that a point \(w\) belongs to \(A^{-}\) if and ondy if \(w\) is either an interior point or a boundary point of \(A\).
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