Metric Spaces and Their Applications

1. Prove that the theorem on shrinking balls is valid in a complete space $X$.

2. Prove that if the theorem on shrinking balls holds in a metric space $X$, then it is complete.

Prove that every uniformly continuous function on a metric space $X$ can be uniquely extended to a continuous function on its completion, and that this extension is uniformly continuous.

Prove that the following metric spaces are not complete, and construct their completions:

Define a distance on the set of segments on the line by the formula $d([a,b],[c,d])=\left|a-c\right|+\left|b-d\right|$. Prove that the resulting metric space is not complete, and find its completion.

Define the distance on the set $\left\{ \Delta \right\}$ of segments on the line to be the length of the symmetric difference, i.e.,

Prove that this metric space is not complete and find its completion.

Prove that the space $B(X)$ of all bounded functions on a set $X$ is complete with respect to the distance

Let $X$ be a bounded metric space. Prove that the correspondence $x\mapsto d(x,\cdot )$ is an isometric mapping of $X$ into $B(X)$ (see Problem 23).

Let $X$ be a subset of a complete metric space $Y$. Prove that:

Let $X$ be a complete metric space, and let $Y_i$ be open dense subsets of $X$. Prove that $\bigcap_{i=1}^{\infty }Y_i$ is dense in $X$.

Prove that the space of polynomials is not complete with respect to the distances:

Let the distance on the set $C(X,Y)$ of continuous mappings of a metric space $X$ into a bounded complete metric space $Y$ be defined by the formula

Prove that $C(X,Y)$ is complete.

Let $X$ be a bounded complete metric space with metric $d_X$, and $G$ the collection of all homeomorphisms of $X$ onto itself. Define a distance on $G$ by the formula

Prove that $G$ is a complete metric space.

Let $p$ be a prime number. Define the $p$-adic norm on the set of rational numbers by the equation

where $m$ and $n$ are integers relatively prime to $p$. Prove:

Prove that the set $\mathbb {Q}$ of rational numbers is a metric space when equipped with the distance $d_p(r_1,r_2)=\left\| r_1-r_2\right\|_{p}$ (see Problem 30). Let $\mathbb {Q}_p$ be its completion. Prove that all the arithmetical operations in $\mathbb {Q}$ extend by continuity to $\mathbb {Q}_p$. The field so obtained is called the field of $p$-adic numbers (regarding the noncompleteness of $\mathbb {Q}$ see Problem 32).

Prove that each element of the field $\mathbb {Q}_p$ (see Problem 31) can be uniquely expressed as a $p$-ary fraction of the form

where $0\leq a_i\leq p-1$; there are only finitely many symbols after the separation point but infinitely many before it. In other words, each element $x\in \mathbb {Q}_p$ is the sum of a convergent series $\sum_{i=-k}^{+\infty }a_ip^i$. Prove that $\mathbb {Q}_p\ne \mathbb {Q}$.

Let $\mathbb {Z}_p$ be the closure of the ring of integers $\mathbb {Z}$ in $\mathbb {Q}_p$ (the set of $p$-adic integers). Prove that $\mathbb {Z}_p$ is a compact set. Construct a homeomorphism of $\mathbb {Z}_p$ onto the Cantor set.

Prove that $\lim_{n\to \infty }x^{p^n}$ exists for any $x\in \mathbb {Z}_p$ (see Problem 33). Denote this limit by $\operatorname {sgn}_p x$. Prove that the resulting function (the $p$-adic signum) takes exactly $p$ distinct values: $0$ and the $(p-1)$-roots of $1$, of which there are $p-1$. Prove the identity

Define a distance on the set $\mathbb {N}$ of positive integers by setting $d(m,n)=1/k$ if the last $k$ places in the decimal expressions for the numbers $m$ and $n$ coincide.