Sequences in metric spaces. Completeness

Describe all fundamental sequences in a discrete metric space and in the space $\mathbb {N}$.

Prove that in the space $B_{0}$ the limit of the sequence $\left\{ x_{n} = \left(x_{n}^{1}, x_{n}^{2}, \ldots \right)\right\}_{1}^{\infty }$ equals $x = \left(x^{1}, x^{2}, \ldots \right)$ if and only if for every $k \in \mathbb {N}$ there exists $N_{k} \in \mathbb {N}$ such that $x_{n}^{k} = x^{k}$ for all $n > N_{k}$.

Let $X$ be a metric space, $A$ its subset, $x$ an arbitrary point of $X$. Prove that $\operatorname {dist}(x, A) = \operatorname {dist}(x, \bar{A})$.

Let $(X, \rho )$ be a metric space, and let the function $\varphi (\xi )$ be continuous and concave (not necessarily strictly) on $[0,+\infty )$, with $\varphi (\xi ) > 0$ on $(0,+\infty )$ and $\varphi (0) = 0$. Prove that $\rho_{1}(x, y) = \varphi (\rho (x, y))$ defines a metric on $X$, with $\rho_{1} \sim \rho$, and if the function $\varphi$ is moreover bounded on $[0,+\infty )$, then the new metric space $(X, \rho_{1})$ is also bounded.

Prove that the metric space $\mathbb {Q}$ with the usual metric $\rho (x, y) = \left|x-y\right|$ is not complete. Does there exist a countable complete metric space?

Let an incomplete metric space $(X, \rho )$ be dense in a metric space $(Y, \rho )$. Prove that if every fundamental sequence of points of the set $X$ has a limit in the space $Y$, then $(Y, \rho )$ is a completion of the space $(X, \rho )$.

Prove that the metric space $\mathbb {R}$ is a completion of the space $\mathbb {Q}$.

For each of the following functions $\rho_{k}: \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}$, determine whether it satisfies the axioms of a metric, and if it does, whether it is equivalent to the standard metric $\rho_{0}(x, y) = \left|x-y\right|$. Is the space $\mathbb {R}$ complete with respect to the metric $\rho_{k}$? For incomplete spaces find the completion.

Which of the following metric spaces are complete (for incomplete spaces it is necessary to describe their completion):

Prove that in the space $\mathbb {Q}_{p}$ the sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ is fundamental if and only if $\rho_{p}\left(x_{n}, x_{n+1}\right) \rightarrow 0$.

Prove that the space $\overline{\mathbb {Q}}_{p}$ (see the list of spaces) is indeed a completion of the space $\mathbb {Q}_{p}$.

Prove that if a metric space $X$ is complete, then its completion $Y$ is isomorphic to the space $X$.

Let there be given a metric space $(X, \rho ) = \left(X_{1}, \rho_{1}\right) \times \left(X_{2}, \rho_{2}\right)$, where $\rho = \rho_{1}+\rho_{2}$, and $X_{1}$ and $X_{2}$ are complete metric spaces. Prove that $X$ is complete.

Let $(X, \rho )$ be a complete metric space, $Y \subset X$. Prove that the space $(Y, \rho )$ is complete if and only if the set $Y$ is closed in $X$.

Give an example of complete metric spaces $\left(X, \rho_{1}\right)$ and $(Y, \rho_{2})$ such that $Y \subset X$, but $Y$ is not closed in $(X, \rho_{1})$. Give an example of a complete metric space $(X, \rho_{1})$ and an incomplete metric space $(Y, \rho_{2})$, for which $Y \subset X$ and $Y$ is closed in $(X, \rho_{1})$.

Give an example of a complete metric space in which the distance from a point to a closed set may not be attained.

Prove Theorem 1.2.

Theorem 1.2 (on nested balls): A metric space is complete if and only if every sequence of closed nested balls whose radii tend to zero has a common point.

Give an example of a complete metric space in which there is a sequence of nested closed balls $B_{1} \supset B_{2} \supset \ldots$, whose radii tend to a positive number, and $\bigcap_{n = 1} B_{n} = \varnothing$.

Give counterexamples to the theorem on nested balls for the case when the metric space is incomplete, and for the case when the balls are open.

Prove that in a complete metric space any sequence of nested nonempty closed bounded sets with diameters tending to zero has exactly one point in its intersection.

Prove that there do not exist countable complete metric spaces without isolated points.