Math Problems Vault

Problem 13.7

Show that the following induce the same convergent sequences in Euclidean space:

\begin{aligned} & \rho ^{*}(x, y) = \left|x_{1}-y_{1}\right|+\cdots +\left|x_{n}-y_{n}\right| \\ & \rho ^{+}(x, y) = \max \left\{ \left|x_{1}-y_{1}\right|, \ldots ,\left|x_{n}-y_{n}\right|\right\} . \end{aligned}

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Problem 13.8

Find a metric on $\mathbb {R}^{n}$ that does not induce the same convergent sequences as the metric induced by the Euclidean norm.

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Problem 13.9

Show that in a metric space an open ball is open.

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Problem 13.10

In a metric space $X$ , is it possible for $r > 0$ and two distinct points $u$ and $v$ in $X$ to have $B(u, r) = B(v, r)$ ? Is this possible in Euclidean space $\mathbb {R}^{n}$ ? Is it possible in a normed linear space?

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Problem 13.11

Let $(X, \rho )$ be a metric space in which $\left\{ u_{n}\right\} \rightarrow u$ and $\left\{ v_{n}\right\} \rightarrow v$ . Show that $\left\{ \rho \left(u_{n}, v_{n}\right)\right\} \rightarrow \rho (u, v)$ .

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Problem 13.12

(i)(ii)

Let $X$ be a metric space, $x$ belong to $X$ and $r > 0$ .

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(i)

Show that $\bar{B}(x, r)$ is closed and contains $B(x, r)$ .

(ii)

Show that in a normed linear space $X$ the closed ball $\bar{B}(x, r)$ is the closure of the open ball $B(x, r)$ , but this is not so in a general metric space.

Problem 13.13

Prove Proposition 2.

Proposition 2: Let $X$ be a subspace of the metric space $Y$ and $E$ a subset of $X$ . Then $E$ is open in $X$ if and only if $E = X \cap O$ , where $O$ is open in $Y$ .

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Problem 13.14

Let $X$ be a subspace of the metric space $Y$ and $A$ a subset of $X$ . Show that $A$ is closed in $X$ if and only if $A = X \cap F$ , where $F$ is closed in $Y$ .

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Problem 13.15

(i)(ii)

Let $X$ be a subspace of the metric space $Y$ .

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(i)

If $\mathcal{O}$ is an open subset of the metric subspace $X$ , is $\mathcal{O}$ an open subset of $Y$ ? What if $X$ is an open subset of $Y$ ?

(ii)

If $F$ is a closed subset of the metric subspace $X$ , is $F$ a closed subset of $Y$ ? What if $X$ is a closed subset of $Y$ ?

Problem 13.16

For a subset $E$ of a metric space $X$ , a point $x \in X$ is called an interior point of $E$ provided that there is an open ball centered at $x$ that is contained in $E$ : the collection of interior points of $E$ is called the interior of $E$ and denoted by int $E$ . Show that int $E$ is always open and $E$ is open if and only if $E = \operatorname {int} E$ .

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Problem 13.17

For a subset $E$ of a metric space $X$ , a point $x \in X$ is called an exterior point of $E$ provided that there is an open ball centered at $x$ that is contained in $X \sim E$ : the collection of exterior points of $E$ is called the exterior of $E$ and denoted by ext $E$ . Show that ext $E$ is always open. Show that $E$ is closed if and only if $X \sim E = \operatorname {ext} E$ .

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Problem 13.18

(i)(ii)(iii)

For a subset $E$ of a metric space $X$ , a point $x \in X$ is called a boundary point of $E$ provided that every open ball centered at $x$ contains points in $E$ and points in $X \sim E$ : the collection of boundary points of $E$ is called the boundary of $E$ and denoted by bd $E$ . Show

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(i)

that $\operatorname {bd} E$ is always closed,

(ii)

that $E$ is open if and only if $E \cap \operatorname {bd} E = \emptyset$ ,

(iii)

that $E$ is closed if and only if $\operatorname {bd} E \subseteq E$ .

Problem 13.19

Let $A$ and $B$ be subsets of a metric space $X$ . Show that if $A \subseteq B$ , then $\bar{A} \subseteq \bar{B}$ . Also, show that $(\overline{A \cup B}) = \bar{A} \cup \bar{B}$ and $(\overline{A \cap B}) \subseteq \bar{A} \cap \bar{B}$ .

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Problem 13.20

Show that for a subset $E$ of a metric space $X$ , the closure of $E$ is the intersection of all closed subsets of $X$ that contain $E$ .

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Problem 13.21

Let $\rho$ be a metric on a set $X$ . Define

\tau (u, v) = \frac{\rho (u, v)}{1+\rho (u, v)} \text{ for all } u, v \in X

Verify that $\tau$ is a bounded metric on $X$ and convergence of sequences with respect to the $\rho$ metric and the $\tau$ metric is the same. Conclude that sets that are closed with respect to the $\rho$ metric are closed with respect to the $\tau$ metric and that sets that are open with respect to the $\rho$ metric are open with respect to the $\tau$ metric.

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