Determine, with proof, which of the following sets \(A\) is a neighborhood of \(a\) in the indicated metric space.
(a)
\(A = \{(x,y) \in \R^2 \mid x^2+y^2 \lt 1\}\) in \((\R^2,d_E)\) with \(a = (0.5,0.5)\)
(b)
\(A\) is the \(x\)-axis in \((\R^2,d_T)\) with \(a =(0,0)\text{,}\) where \(d_T\) is the taxicab metric
(c)
\(A\) is the set of rational numbers in \((\R, d_E)\) with \(a = 0\)
(d)
\(A\) is the set of positive integers in \((Q,d)\) and \(a = 1\text{,}\) where \(Q\) is the set of all rational numbers in reduced form with metric \(d : Q \times Q \to \R\) defined by
(The fact that \(d\) is a metric is the topic of Exercise 3.)
2.
Let \(X = \{1,3,5\}\) and define \(d_X: X \times X \to \R\) by \(d_X(x,y) = xy - 1 \pmod{8}\text{.}\) That is, \(d_X(x,y)\) is the remainder when \(xy - 1\) is divided by \(8\text{.}\) That \(d_X\) is a metric on \(X\) is examined in Exercise 2. Let \((Y,d_Y)\) be a metric space. Is it possible to define a function \(f: X \to Y\) that is not continuous? Explain.
3.
If \(x = (x_1, x_2, \ldots, x_n)\text{,}\) we let \(|x| = \sqrt{x_1^2+x_2^2+ \cdots + x_n^2}\text{.}\) For \(x = (x_1, x_2, \ldots,
x_n)\) and \(y = (y_1, y_2, \ldots y_n)\text{,}\) define \(d_H: \R^n \times \R^n \to \R\) by
The fact that \(d_H\) is a metric is examined in Exercise 7. Let \((X,d_X) = (\R^2, d_H)\) and let \((Y,d_Y) = (\R, d_E)\text{.}\) Define \(f: X \to Y\) and \(g: X \to Y\) by
\begin{equation*}
f(x) = \begin{cases}0 \amp \text{ if } x = (0,0) \\ 1 \amp \text{ otherwise } \end{cases} \ \text{ and } \ g(x) = \begin{cases}0 \amp \text{ if } |x|\lt 1 \\ 1 \amp \text{ otherwise } \end{cases}\text{.}
\end{equation*}
One of \(f\text{,}\)\(g\) is continuous and the other is not. Determine which is which, with proof for each.
4.
Recall from Chapter 3 that we can construct a finite metric space by starting with a finite set of points and making a graph with the points as vertices. We construct edges so that the graph is connected (that is, there is a path from any one vertex to any other) and give weights to the edges. We then define a metric \(d\) on \(S\) by letting \(d(x,y)\) be the length of a shortest path between vertices \(x\) and \(y\) in the graph. Consider the metric space \((X,d)\) corresponding to the graph in Figure 7.7.
Figure7.7.A graph to define a metric.
(a)
Determine all of the open balls \(B(a,\delta)\) for every positive real number \(\delta\text{.}\)
(b)
Find all of the neighborhoods of \(a\text{.}\)
5.
(a)
Let \(f: (\R,d_E) \to (\R,d_E)\) be defined by \(f(x) = ax+b\) for some real numbers \(a\) and \(b\) with \(a \neq 0\text{.}\) Let \(p \in \R\) and let \(r \gt 0\text{.}\) Show that \(f^{-1}(B(f(p),r))\) contains an open ball centered at \(p\text{.}\) Conclude that every linear function from \((\R,d_E)\) to \((\R,d_E)\) is continuous.
By Exercise 5 we can assume \(a \gt 0\) to simplify the problem.
(b)
Let \(f: (\R,d_E) \to (\R,d_E)\) be defined by \(f(x) = ax^2+bx+c\) for some real numbers \(a\text{,}\)\(b\text{,}\) and \(c\) with \(a \neq 0\text{.}\) Let \(p \in \R\) and let \(r \gt 0\text{.}\)\(f^{-1}(B(f(p),r))\) contains an open ball centered at \(p\text{.}\) Conclude that every quadratic function from \((\R,d_E)\) to \((\R,d_E)\) is continuous.
for all \(b, c \in X\text{.}\) Define \(f: X \to \R\) by \(f(x) = d(x,A)\text{.}\) Let \(b \in X\text{.}\) Given \(\epsilon \gt 0\text{,}\) show that there is a neighborhood \(N\) of \(b\) such that \(x \in N\) implies \(f(x) \in B(f(b),\epsilon)\text{.}\) Conclude that \(f\) is a continuous function. (Assume the metric on \(\R\) is the Euclidean metric.)
7.
Let \(a\) and \(b\) be distinct points of a metric space \(X\text{.}\) Prove that there are neighborhoods \(N_a\) and \(N_b\) of \(a\) and \(b\) respectively such that \(N_a \cap N_b = \emptyset\text{.}\)
8.
Let \((X,d)\) be a metric space and let \(a \in X\text{.}\) Prove each of the following.
(a)
There is a neighborhood that contains \(a\text{.}\)
(b)
If \(N\) is a neighborhood of \(a\) and \(N \subseteq M\text{,}\) then \(M\) is a neighborhood of \(a\text{.}\)
(c)
If \(M\) and \(N\) are neighborhoods of \(a\text{,}\) then so is \(M \cap N\text{.}\)
9.
Let \(f: (\R,d_E) \to (\R,d_E)\) be a continuous function. Show that if \(f(a) \gt 0\) for some \(a \in \R\text{,}\) then there is a neighborhood \(N\) of \(a\) such that \(f(x) \gt 0\) for all \(x \in N\text{.}\)
10.
Let \((X,d)\) be a metric space where \(d\) is the discrete metric. Show that every subset of \(X\) is a neighborhood of each of its points.
11.
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 \(N\) is a neighborhood of a point \(a\) in a metric space \(X\text{,}\) then any open ball contained in \(N\) is also a neighborhood of \(a\text{.}\)
(b)
If \(N\) is a neighborhood of a point \(a\) in a metric space \(X\text{,}\) then \(N\) is a neighborhood of each of its points.
(c)
If \(X\) and \(Y\) are metric spaces and \(f : X \to Y\) is a continuous function, then \(f(N)\) is a neighborhood of \(f(a)\) in \(Y\) whenever \(N\) is a neighborhood of \(a\) in \(X\text{.}\)
(d)
If \(X\) and \(Y\) are metric spaces and \(f : X \to Y\) is continuous at \(a \in X\text{,}\) and \(N\) is a neighborhood of \(f(a)\) in \(Y\text{,}\) then \(f^{-1}(N)\) is a neighborhood of \(a\) in \(X\text{.}\)
(e)
If \(a\) is a point in a metric space \(X\) and if \(\delta\) is a positive real number, then the open ball \(B(a,\delta)\) contains infinitely many points of \(X\text{.}\)
(f)
If \(N_1\text{,}\)\(N_2\text{,}\)\(\ldots\text{,}\)\(N_k\) are neighborhoods of a point \(a\) in a metric space \(X\) for some positive integer \(k\text{,}\) then \(\bigcap_{i=1}^k N_i\) is a neighborhood of \(a\text{.}\)
(g)
If \(N_{\alpha}\) is a neighborhood of a point \(a\) in a metric space \(X\) for all \(\alpha\) in some indexing set \(I\text{,}\) then \(\bigcap_{\alpha \in I} N_{\alpha}\) is a neighborhood of \(a\text{.}\)
