Let \(d\) be the discrete metric. Let \((X,d)\) be a metric space.
(a)
Show that every subset of \(X\) is open.
(b)
Let \((Y, d_Y)\) be a metric space. Prove that every function \(f: X \to Y\) is continuous.
(c)
Is it also true that every function \(f: Y \to X\) is continuous? If yes, prove your answer. If no, find a counterexample.
2.
Prove that a subset \(O\) of a metric space \(X\) is open if and only if \(O = \Int(O)\text{.}\)
3.
Let \(A\) and \(B\) be subsets of a metric space \(X\) with \(A \subseteq B\text{.}\) Prove or disprove the following.
(a)
\(\Int(A) \subseteq \Int(B)\)
(b)
\(\Int(\Int(A)) = \Int(A)\)
4.
Let \(a, b \in \R\) with \(a \lt b\text{.}\) Show that the set \((a,b]\) in \((\R, d_E)\) is not an open set.
5.
Let \(A = \{(x,y) \in \R^2 \mid 1 \lt x \lt 3, 0 \lt y \lt 1\}\text{.}\)
(a)
Is \(A\) an open set in \((\R^2, d_E)\text{?}\) Prove your answer.
(b)
Is \(A\) an open set in \((\R^2, d_T)\text{?}\) Prove your answer.
(c)
Is \(A\) an open set in \((\R^2, d_M)\text{?}\) Prove your answer.
6.
Let \(S\) be a finite set of points in \(\R^2\text{.}\) Is the set \(\R^2 \setminus A\) an open set in \((\R^2, d_E)\text{?}\) Prove your answer.
7.
Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces and let \(f: X \to Y\) be a function. Prove that \(f\) is continuous if and only if \(f^{-1}(\Int(B)) \subseteq \Int(f^{-1}(B))\) for every subset \(B\) of \(Y\text{.}\)
8.
Consider the metric space \((Q,d)\text{,}\) where \(d : Q \times Q \to \R\) is defined by
(The fact that \(d\) is a metric is the topic of Exercise 3.) Describe the open ball \(B(q,2)\) in \(Q\) if \(q = \frac{2}{5}\text{.}\)
9.
Let \((X,d)\) be a metric space and let \(x_1\) and \(x_2\) be distinct points in \(X\text{.}\) Prove that there are open sets \(O_1\) containing \(x_1\) and \(O_2\) containing \(x_2\) such that \(O_1 \cap O_2 = \emptyset\text{.}\) (This shows that we can separate points in metric spaces with open sets. Separation properties are important in topology.)
10.
Let \(X = \R\) with the Euclidean metric \(d_E\text{,}\) and let \(Y = \R\) with the metric \(d\) defined by \(d(x,y) = \frac{|x-y|}{|x-y|+1}\) (that \(d\) is a metric is the subject of Exercise 11). Let \(a\) and \(b\) be real numbers and let \(f:\R \to \R\) be defined by \(f(x) = ax+b\text{.}\) That is, \(f\) is an arbitrary linear function from \(\R\) to \(\R\text{.}\)
(a)
Describe the open balls in \((Y, d)\text{.}\) That is, if \(a\) is a real number and \(\delta\) is a positive real number, what is the specific set of points in \(B(a, \delta)\) in \(Y\text{?}\)
(b)
Is \(f\) from \(X\) to \(Y\) continuous for any real numbers \(a\) and \(b\text{?}\) Prove your answer.
(c)
Is \(f\) from \(Y\) to \(X\) continuous for any real numbers \(a\) and \(b\text{?}\) Prove your answer.
11.
Let \(f: \R^2 \to \R\) be defined by \(f((x,y)) = x\text{.}\) Assume that we use the max metric \(d_M\) on \(\R^2\) and the Euclidean metric \(d_E\) on \(\R\text{.}\) Use Theorem to determine if \(f\) a continuous function. Prove your conjecture.
12.
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 \(A\) and \(B\) are nonempty subsets of a metric space \(X\text{,}\) then \(\Int(A \cup B) \subseteq \Int(A) \cup \Int(B)\text{.}\)
(b)
If \(A\) and \(B\) are nonempty subsets of a metric space \(X\text{,}\) then \(\Int(A) \cup \Int(B) \subseteq \Int(A \cup B)\text{.}\)
(c)
If \(A\) and \(B\) are nonempty subsets of a metric space \(X\text{,}\) then \(\Int(A \cap B) \subseteq \Int(A) \cap \Int(B)\text{.}\)
(d)
If \(A\) and \(B\) are nonempty subsets of a metric space \(X\text{,}\) then \(\Int(A) \cap \Int(B) \subseteq \Int(A \cap B)\text{.}\)
(e)
Every subset of an open set in a metric space \((X,d)\) is open in \(X\text{.}\)
(f)
A subset \(O\) of \(\R^2\) is open under the Euclidean metric \(d_E\) if and only if \(O\) is open under the taxicab metric \(d_T\text{.}\)
(g)
Let X = \([0,1] \cup [2,3]\) endowed with the Euclidean metric. Then \([0, 1]\) is an open subset of \(X\text{.}\)
