Metric Spaces: General Properties

The Nikodym Metric. Let ($X, \mathcal{M}, \mu$) be a finite measure space. For $A, B \in \mathcal{M}$, define $\rho (A, B) = m(A \Delta B)$, where $A \Delta B = [A \sim B] \cup [B \sim A]$, the symmetric difference of $A$ and $B$. Define two measurable sets to be equivalent provided that their symmetric difference has measure zero. Show that $\rho$ induces a metric on the collection of equivalence classes. Finally, show that for $A, B \in \mathcal{M}$,

where $\chi_{A}$ and $\chi_{B}$ are the characteristic functions of $A$ and $B$, respectively.

Show that for $a, b, c \geq 0$,

Let ($X, \mathcal{M}, \mu$) be a finite measure space and $\mathcal{F}$ be the collection of measurable real-valued functions on $X$. For $f, g \in \mathcal{F}$, define

Define two measurable functions to be equivalent provided that they are equal almost everywhere on $X$. Show that $\rho$ induces a metric on the collection of equivalence classes.

For $0 < p < 1$, show that

Let ($X, \mathcal{M}, \mu$) be measure space and $0 < p < 1$. For measurable functions $g: X \rightarrow \mathbb {R}$ and $h: X \rightarrow \mathbb {R}$ that have integrable $p$-th powers, define

Define two such functions to be equivalent provided that they are equal almost everywhere on $X$. Show that $\rho_{p}(\cdot , \cdot )$ induces a metric on the collection of equivalence classes.

Show that the triangle inequality for Euclidean space $\mathbb {R}^{n}$ follows from the triangle inequality for $L^{2}[0,1]$.

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

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

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

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?

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

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

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 $Y$.

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 $Y$.

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

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 $E = \operatorname {int} E$.

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 $X \sim E = \operatorname {ext} E$.

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

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 $(\overline{A \cap B}) \subseteq \bar{A} \cap \bar{B}$.

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 $E$.

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

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.

Exhibit a continuous mapping that is not uniformly continuous and a uniformly continuous mapping that is not Lipschitz.

Show that every mapping from a metric space ($X, \rho$) to a metric space ($Y, \sigma$) is continuous if $\rho$ is the discrete metric.

Suppose there is a continuous, one-to-one mapping from a metric space ($X, \rho$) to a metric space ($Y, \sigma$), where $\sigma$ is the discrete metric. Show that every subset of $X$ is open.

For a metric space ($X, \rho$), show that the metric $\rho : X \times X \rightarrow \mathbb {R}$ is continuous, where $X \times X$ has the product metric.

Let $z$ be a point in the metric space ($X, \rho$). Define the function $f: X \rightarrow \mathbb {R}$ by $f(x) = \rho (x, z)$. Show that $f$ is uniformly continuous.

For a non-empty subset $E$ of the metric space ($X, \rho$) and a point $x \in X$, define the distance from $x$ to $E, \operatorname {dist}_{E}(x)$, as follows:

Show that a subset $E$ of a metric space $X$ is open if and only if there is a continuous real-valued function $f$ on $X$ for which $E = \left\{ x \in X \mid f(x) > 0\right\}$.

Show that a subset $E$ of a metric space $X$ is closed if and only if there is a continuous real-valued function $f$ on $X$ for which $E = f^{-1}(0)$.

Let $X = C[a, b]$. Define the function $\psi : X \rightarrow \mathbb {R}$ by

Show that $\psi$ is Lipschitz on the metric space $X$, where $X$ has the metric induced by the maximum norm.

In a metric space $X$, show

In a metric space $X$, show that a Cauchy sequence converges if and only if it has a convergent subsequence.

Define the concept of rapidly Cauchy sequence in a metric space, and show that a space is complete if and only if every rapidly Cauchy sequence converges.

Provide an example of a descending, countable collection of closed, non-empty sets of real numbers for which the intersection is empty. Does this contradict the Cantor Intersection Theorem?

For a mapping $f$ of the metric space ($X, \rho$) to the metric space ($Y, \sigma$), show that $f$ is uniformly continuous if and only if for any two sequences $\left\{ u_{n}\right\}$ and $\left\{ v_{n}\right\}$ in $X$,

For the countable Cartesian product of metric spaces defined in Proposition 7, show that the product is complete if and only if each factor space is complete.

Proposition 7: Let $X = \prod_{n=1}^{\infty } X_{n}$ be the Cartesian product of the countable collection of metric spaces $\left\{ (X_{n}, \rho_{n})\right\}_{n=1}^{\infty }$. For points $u = \left\{ u_{n}\right\}$ and $v = \left\{ v_{n}\right\}$ in $X$, define $\rho (u, v) = \sum_{n=1}^{\infty } \frac{1}{2^{n}} \cdot \frac{\rho_{n}(u_{n}, v_{n})}{1 + \rho_{n}(u_{n}, v_{n})}$. Then $\rho$ is a metric on $X$ for which sequential convergence is componentwise convergence in each component metric space $(X_{n}, \rho_{n})$.

For each index $n$, define $f_{n}(x) = \alpha x^{n}+\beta \cos (x / n)$ for $0 \leq x \leq 1$. For what values of the parameters $\alpha$ and $\beta$ is the sequence $\left\{ f_{n}\right\}$ a Cauchy sequence in the metric space $C[0,1]$ ?

Let $\mathcal{D}$ be the subspace of $C[0,1]$ consisting of the continuous functions $f:[0,1] \rightarrow \mathbb {R}$ that are differentiable on $(0,1)$. Is $\mathcal{D}$ complete?

Define $\mathcal{L}$ to be the subspace of $C[0,1]$ consisting of the functions $f:[0,1] \rightarrow \mathbb {R}$ that are Lipschitz. Is $\mathcal{L}$ complete?

For a metric space ($X, \rho$), complete the following outline of a proof of Theorem 12:

Theorem 12: Let $(X, \rho )$ be a metric space. Then there is a complete metric space $(\widehat{X}, \widehat{\rho })$ for which $X$ is a dense subset of $\widehat{X}$ and $\widehat{\rho }(u,v) = \rho (u,v)$ for all $u, v \in X$. (This space is called the completion of $(X,\rho )$; it is unique up to an isometry that is the identity on $X$.)

Consider the metric space $\mathbb {Q}$ consisting of the rational numbers with the metric induced by the absolute value. Which subspaces of $\mathbb {Q}$ are complete and which are compact?

Let $B = B(x, r)$ be an open ball in Euclidean space $\mathbb {R}^{n}$. Show that $B$ fails to be compact by

When is a set $X$ with the discrete metric a compact metric space?

Prove Proposition 23.

Proposition 23: If a subspace $K$ of a metric space $X$ is compact, then $K$ is a closed, bounded subset of $X$.

Prove that if $X$ is compact, then $K \subseteq X$ is compact if and only if it is closed subset of $X$.

For the countable Cartesian product of metric spaces considered in Proposition 7, show that the product is compact if and only if each factor space is compact.

Proposition 7: Let $X = \prod_{n=1}^{\infty } X_{n}$ be the Cartesian product of the countable collection of metric spaces $\left\{ (X_{n}, \rho_{n})\right\}_{n=1}^{\infty }$. For points $u = \left\{ u_{n}\right\}$ and $v = \left\{ v_{n}\right\}$ in $X$, define $\rho (u, v) = \sum_{n=1}^{\infty } \frac{1}{2^{n}} \cdot \frac{\rho_{n}(u_{n}, v_{n})}{1 + \rho_{n}(u_{n}, v_{n})}$. Then $\rho$ is a metric on $X$ for which sequential convergence is componentwise convergence in each component metric space $(X_{n}, \rho_{n})$.

Let $E$ be a subset of the compact metric space $X$. Show the metric subspace $E$ is compact if and only if $E$ is a closed subset of $X$.

Let $\left\{ F_{n}\right\}_{n = 1}^{\infty }$ be a descending countable collection of non-empty closed subsets of a compact metric space $X$. Show that $\bigcap_{n = 1}^{\infty } F_{n} \neq \emptyset$.

For a subset $E$ of a metric space $X$, show that $E$ is totally bounded if and only if its closure $\bar{E}$ is totally bounded.

For a subset $E$ of a complete metric space $X$, show that $E$ is totally bounded if and only if its closure $\bar{E}$ is compact.

Let $B = \left\{ \left\{ x_{n}\right\} \in \ell^{2} \mid \sum_{n = 1}^{\infty } x_{n}^{2} \leq 1\right\}$ be the closed unit ball in $\ell^{2}$. Show that $B$ fails to be compact by

Let $B = \left\{ f \in L^{2}[a, b] \mid \left\| f\right\|_{2} \leq 1\right\}$ be the closed unit ball in $L^{2}[a, b]$. Show that $B$ fails to be compact by

Let $X$ be a totally bounded metric space.

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

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.

Let $E$ be a subset of Euclidean space $\mathbb {R}^{n}$. Assume that every continuous real-valued function on $E$ takes a minimum value. Is $E$ necessarily closed and bounded?

Suppose $f$ is a continuous real-valued function on Euclidean space $\mathbb {R}^{n}$ with the property that there is a positive number $c$ such that $\left|f(x)\right| \geq c \cdot \left\| x\right\|$ for all $x \in \mathbb {R}^{n}$. Show that if $K$ is a compact set of real numbers, then its inverse image under $f, f^{-1}(K)$, also is compact. (Mappings with this property are called proper.)

For a compact metric space ($X, \rho$), show that there are points $u, v \in X$ for which $\rho (u, v) = \operatorname {diam} X$.

Let $K$ be a compact subset of the metric space ($X, \rho$) and $x_{0}$ belong to $X$. Show that there is a point $z \in K$ for which

Let $K$ be a compact subset of the metric space $X$. For a point $x \in X \sim K$, show that there is an open set $\mathcal{U}$ containing $K$ and an open set $\mathcal{O}$ containing $x$ for which $\mathcal{U} \cap \mathcal{O} = \emptyset$.

Let $A$ and $B$ be subsets of a metric space ($X, \rho$). Define

If $A$ is compact and $B$ is closed, show that $A \cap B = \emptyset$ if and only if $\operatorname {dist}(A, B) > 0$.

Let $K$ be a compact subset of a metric space $X$ and $\mathcal{O}$ an open set containing $K$. Use the preceding problem to show that there is an open set $\mathcal{U}$ for which $K \subseteq \mathcal{U} \subseteq \overline{\mathcal{U}} \subseteq \mathcal{O}$.

Let $X$ be a metric space that contains a finite dense subset $D$. Show that $X = D$.

Show that if two continuous mappings defined on a metric space $X$ take the same values on a dense subset, then they are equal.

Prove Lusin's Theorem for a finite Borel measure space ($X, \mathcal{B}(X), \mu$) where $X$ is a separable, complete metric space.