Informal, but convincing, arguments suffice for this problem.
(a)
Let \(D = \{(x,y) \in \R^2 \mid x^2+y^2 \leq 1\}\) as a subset of \((\R^2,d_E)\text{.}\) Note that \(D\) is the unit disk in the plane. Determine all of the interior points, boundary points, accumulation points, and isolated points of \(D\text{.}\) Give reasons for your conclusions. Is \(D\) an open set? Is \(D\) a closed set? Explain.
(b)
Let \(A = \Q\text{,}\) the set of rational numbers, as a subset of \((\R,d_E)\text{.}\) Determine all of the interior points, boundary points, accumulation points, and isolated points of \(A\text{.}\) Give reasons for your conclusions. Is \(A\) an open set? Is \(A\) a closed set? Explain.
(c)
Let \(A = \left\{\frac{1}{n} \ \left| \right. \in \Z^+\right\}\) as a subset of \((\R,d_E)\text{.}\) Determine all of the interior points, boundary points, accumulation points, and isolated points of \(A\text{.}\) Give reasons for your conclusions. Is \(A\) an open set? Is \(A\) a closed set? Explain.
2.
Let \((X,d)\) be a metric space. Let \(a \in X\text{,}\) and let \(r \gt 0\text{.}\) We know that the open ball \(B(a,r) = \{x \in X \mid d(a,x) \lt r\}\) is an open set. Let
Prove or disprove: \(B[a,r]\) is a closed set in \(X\text{.}\)
3.
Let \((X,d)\) be a metric space. We have seen that it is possible for a subset of \(X\) to be both open and closed. There is a characterization of sets that are both open and closed in terms of their boundaries. Find and prove such a characterization. (Your statement should have the form: A subset \(A\) of a metric space \(X\) is both open and closed if and only if the boundary of \(A\) is \(\underline{}\text{.}\))
4.
Let \(A\) be a subset of a metric space. Let \(A'\) be the set of limit points of \(A\) and \(A^i\) the set of isolated points of \(A\text{.}\) Prove the following.
(a)
\(A \cup A^i = A \cup A'\)
(b)
\(A' \cap A^i = \emptyset\)
(c)
\(A \subseteq A' \cup A^i\)
(d)
\(x \in \overline{A}\) if and only if there is a sequence of points of \(A\) which converges to \(x\)
(e)
\(\overline{A}\) is the intersection of all closed sets that contain \(A\)
(f)
\(\Int(A)\) is the union of all open sets contained in \(A\)
(g)
\(\overline{A}\) is the disjoint union of \(\Int(A)\) and \(\Bdry(A)\)
(h)
\(\overline{X \setminus A} = X \setminus \Int(A)\)
(i)
\(\Int(X \setminus A) = X \setminus \overline{A}\)
5.
Let \((X,d)\) be a metric space and \(A\) a subset of \(X\text{.}\) Prove that a point \(x \in X\) is a limit point of \(A\) if and only if every open ball centered at \(x\) contains a point in \(A\) different from \(x\text{.}\)
6.
Let \(A\) be a subset of a metric space. Let \(A'\) be the set of limit points of \(A\) and \(A^i\) the set of isolated points of \(A\text{.}\)
(a)
Prove that \(A' \cap A^i = \emptyset\) and \(A \subseteq A' \cup A^i\text{.}\)
(b)
Prove that \(x \in \overline{A}\) if and only if there is a sequence of points of \(A\) which converges to \(x\text{.}\)
(c)
Prove that if \(F\) is a closed set such that \(A \subseteq F\text{,}\) then \(\overline{A} \subseteq F\text{.}\) Then prove that \(\overline{A}\) is the intersection of all such closed sets \(F\) and hence is closed.
7.
Prove Theorem 10.7 that if \(A\) is a subset of a metric space \(X\) and \(b\) is a boundary point of \(A\text{,}\) then there are sequences \((x_n)\) in \(X \setminus A\) and \((a_n)\) in \(A\) that converge to \(b\text{.}\)
8.
Recall that the distance from a point \(x\) in a metric space \(X\) to a nonempty subset \(A\) of \(X\) is
\begin{equation*}
d(a,X) = \inf\{d(x,a) \mid a \in A\}\text{.}
\end{equation*}
Prove that a subset \(C\) of a metric space \(X\) is closed if and only if whenever \(x \in X\) and \(d(x,C) = 0\text{,}\) then \(x \in C\text{.}\)
9.
Let \((X,d)\) be a metric space. In this exercise we show that some subsets of \(X\text{,}\) other than \(\emptyset\) and \(X\) must be closed. Show that any finite subset of \(X\) is closed.
Prove that a subset \(C\) of a metric space \(X\) is closed if and only if \(C\) contains its boundary.
11.
Let \((X,d_X)\) and \((Y,d_Y)\) be metric space and let \(f: X \to Y\) be a function.
(a)
Prove that \(f\) is continuous if and only if \(f^{-1}(\Int(B)) \subseteq \Int(f^{-1}(B))\) for any subset \(B\) of \(Y\text{.}\)
(b)
Give an example where the containment, and not the equality, in (a) is the best we can do.
(c)
Give an example to show that the equality in (a) can actually be achieved.
12.
Let \((X,d)\) be a metric space and let \(A\) be a subset of \(X\text{.}\) Prove that every boundary point of \(A\) is either a limit point or an isolated point of \(A\text{.}\)
13.
Let \((X,d)\) be a metric space, and let \(A\) and \(B\) be subsets of \(X\text{.}\)
(a)
Is it the case that \(\overline{A \cup B} = \overline{A} \cup \overline{B}\text{?}\) If true, prove it. If false, show why and prove any containment that is true.
(b)
Is it the case that \(\overline{A \cap B} = \overline{A} \cap \overline{B}\text{?}\) If true, prove it. If false, show why and prove any containment that is true.
14.
Recall that an infinite union of closed sets in a metric space may not be closed, and that an infinite intersection of open sets in a metric space may not be open. In this exercise we explore situations in which we can conclude that an infinite union of closed sets is closed and an infinite intersection of open sets is open. Let \((X,d)\) be a metric space.
(a)
We first establish a preliminary result. Let \(C\) be a closed subset of \(X\) and \(x \in X\text{.}\) Prove that if \(x \notin C\text{,}\) then \(d(x,C) \gt 0\text{.}\)
(b)
Let \(\{C_{\alpha}\}\) be a collection of closed subsets of \(X\) for \(\alpha\) in some indexing set \(I\) with the property that given any \(x \in X\text{,}\) there exists an \(\epsilon_x \gt 0\) such that \(B(x, \epsilon_x)\) intersects at most finitely many of the sets \(C_{\alpha}\text{.}\) Prove that \(\bigcup_{\alpha \in I} C_{\alpha}\) is closed.
(c)
Determine and prove an analogous statement for open sets in \(X\text{.}\)
15.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
(a)
If \(x\) is a point in a metric space \(X\text{,}\) then the singleton set \(\{x\}\) is closed.
(b)
The only subsets of \(\R\) that are both open and closed under the standard metric are \(\emptyset\) and \(\R\text{.}\)
(c)
If \((X,d)\) is the metric space with \(X = \{1,3,5\}\) and \(d(x,y) = xy - 1 \pmod{8}\text{,}\) then the set \(\{1,3\}\) is both open and closed in \(X\text{.}\)
(d)
If \(X\) is a metric space and \(A \subseteq X\text{,}\) then \(\Int(\overline{A}) = A\text{.}\)
(e)
The boundary of any subset of a metric space \(X\) is a closed set.
(f)
If \(A\) is a subset of a metric space \(X\text{,}\) then \(A \subseteq A' \cup A^i\) where \(A'\) is the set of limit points of \(A\) and \(A^i\) is the set of isolated points of \(A\)
