Metric spaces

Verify the axioms of a metric in the following spaces (see the list of spaces):

Verify the axioms of a metric $\rho = \rho_{1}+\rho_{2}$ in the space $(X, \rho ) = \left(X_{1}, \rho_{1}\right) \times \left(X_{2}, \rho_{2}\right)$, where $X_{1}$ and $X_{2}$ are metric spaces.

Prove that in a metric space the following conditions are equivalent:

1. the set is closed;

2. the set contains all its boundary points;

3. the set contains all its limit points.

Is it true that in an arbitrary metric space the closure of the open ball $B(x, r)$ is the closed ball $\bar{B}(x, r)$?

Prove that for an arbitrary set $A$ of a metric space $X$ the following statements hold:

Describe all open and all closed sets in the space $\operatorname {discr}(X)$.

Let $X$ be an arbitrary metric space. Prove that arbitrary unions and finite intersections of open sets are open. Give an example of an infinite system of open sets whose intersection is not open.

Prove that in a metric space $X$ the intersection of any number of open sets is open if and only if $X$ consists of isolated points ¹.

Give an example of a metric space $X$ and two distinct sets $A$ and $B$ in it such that $A \subsetneq B$, but $\bar{A} \supsetneq B$.

Give an example of a metric space and balls $B\left(x, r_{1}\right)$ and $B\left(y, r_{2}\right)$ in it such that $r_{1} > r_{2}$, but $B\left(x, r_{1}\right) \varsubsetneqq B\left(y, r_{2}\right)$.

Prove that in the space $\mathbb {Q}_{p}$ the triangle axiom holds in a strengthened form: $\rho (x, y) \leqslant \max \left\{ \rho (x, z), \rho (y, z)\right\}$, and moreover, among the three numbers $\rho (x, y), \rho (x, z)$ and $\rho (y, z)$ two are necessarily equal (all "triangles" in this space are isosceles).

Prove that in the space $\mathbb {Q}_{p}$:

Prove that:

Let $A$ be a bounded set in a metric space. Prove that the closure $\bar{A}$ is also bounded and $\operatorname {diam}(\bar{A}) = \operatorname {diam}(A)$.

Prove that for any bounded set in a metric space the inequalities hold

Prove that in the space $l_{2}(2)$ for every bounded set $M$ the Chebyshev center exists and $\operatorname {diam}(M) / 2 \leqslant r_{C}(M) \leqslant \operatorname {diam}(M) / \sqrt{3}$.

Prove that in the space $l_{\infty }(2)$ for every bounded set $M$ the Chebyshev center exists and $r_{C}(M) = \operatorname {diam}(M) / 2$.

Give an example of a metric space in which for every bounded set $M$ the equality $r_{C}(M) = \operatorname {diam}(M)$ holds.

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.

Prove that a set $A$ in a metric space $X$ is nowhere dense if and only if the set $\bar{A}$ contains no ball.

Prove that the complement of a nowhere dense set is everywhere dense. Give an example showing that the complement of an everywhere dense set need not be a nowhere dense set. Prove that the complement of an everywhere dense open set is nowhere dense.

Prove that the closure of a nowhere dense set is nowhere dense.

Let the set $U$ be open, and the set $F$ closed in a metric space $X$. Denote by $F^{0}$ the interior of the set $F$. Prove that the following sets are nowhere dense in $X$:

Prove that the intersection of at most countably many open everywhere dense sets in a complete metric space is an everywhere dense set.

Prove Theorem 1.3. Show the essentiality of the completeness condition of the space in the statement of the theorem.

Theorem 1.3 (R. Baire, 1899): A complete metric space cannot be represented as a countable union of nowhere dense sets.

Do there exist incomplete metric spaces for which Baire's theorem holds?

Prove that in a metric space consisting only of isolated points, the only everywhere dense set is the whole space, and the only nowhere dense set is the empty set.

Let, in a metric space $(X, \rho )$, a set $A$ be contained in the closure of a set $B$, where $B$ is countable. Prove that the metric space $(A, \rho )$ is separable. In particular, any subspace of a separable metric space is separable.

Prove that the completion of a separable metric space is separable.

Prove that the following spaces are separable:

Let $(X, \rho )$ be a metric space, and let $\varphi (\xi )$ be a nonnegative, continuous, concave, and bounded function on $[0,+\infty )$, with $\varphi (0) = 0$. Prove that if $(X, \rho )$ is separable, then $(X, \rho_{1})$, where $\rho_{1} = \varphi (\rho )$, is also separable.

Prove that there exists a function, continuous on the segment $[0,1]$, different from a constant, and not monotonic on any interval of this segment.

Prove that there does not exist a function $U(x, y)$, continuous on the square $[0,1] \times [0,1]$, that is universal in the following sense: for every function $f \in C[0,1]$ satisfying $\max_{x \in [0,1]}\left|f(x)\right| \leqslant 1$, there exists a number $y = y(f) \in [0,1]$ such that $f(x) \equiv U(x, y(f))$.

Prove that if a function $f:[0,1] \rightarrow \mathbb {R}$ is a pointwise limit of continuous functions ¹, then its set of points of continuity is everywhere dense on the segment $[0,1]$.

Let the function $f$ be holomorphic in a domain $D$ and for every $z \in D$ there exist an index $n = n(z)$ such that $f^{(n)}(z) = 0$. Prove that $f$ is a polynomial.

Let the function $f$ be infinitely differentiable on the segment $[0,1]$ and for every $x \in (0,1)$ there exist an index $n = n(x)$ such that $f^{(n)}(x) = 0$. Prove that $f$ is a polynomial.

Let two metrics $\rho_{1}$ and $\rho_{2}$ be introduced on a set $X$. Prove that $\rho_{1} \prec \rho_{2}$ if and only if the mapping $f:\left(X, \rho_{2}\right) \rightarrow \left(X, \rho_{1}\right), f(x) = x$, is continuous. Prove that $\rho_{1} \sim \rho_{2}$ if and only if $f$ is a homeomorphism.

Let $X$ be a metric space, $M \subset X$ an arbitrary nonempty set. Prove that the mapping $f: x \mapsto \operatorname {dist}(x, M)$ from $X$ to $\mathbb {R}$ is continuous.

Let $X$ and $Y$ be metric spaces, and $f: X \rightarrow Y$ a continuous surjective mapping. Prove that if $M$ is an everywhere dense set in $X$, then its image $f(M)$ is an everywhere dense set in $Y$.

Give an example of a function $f: X \rightarrow \mathbb {R}$, continuous but not bounded on some ball, where $X$ is a complete metric space.

In complete metric spaces give examples of a locally uniformly continuous mapping that is not uniformly continuous, and of a continuous mapping that is not locally uniformly continuous.

Let $X = \operatorname {discr}(M)$. Prove that an isometric embedding $f: X \rightarrow \mathbb {R}^{n}$ exists if and only if the space $X$ contains at most $n+1$ points.

Prove that the metric spaces $\mathbb {R}$ and the real space $l_{2}(2)$ are not homeomorphic.

Prove that the real metric spaces $l_{2}(2)$ and $l_{\infty }(2)$ are homeomorphic, but not isometric.

Let the function $f: \mathbb {R} \rightarrow \mathbb {R}$ be differentiable on $\mathbb {R}$ and satisfy $\sup_{t \in \mathbb {R}}\left|f^{\prime }(t)\right| \leqslant q < 1$. Prove that the equation $f(t) = t$ has a unique solution on $\mathbb {R}$.

Using the contraction mapping theorem, compute the limit of the sequence of continued fractions: $2 ; 2+\frac{1}{2}$; $2+\frac{1}{2+\frac{1}{2}} ; \ldots$

Let the function $f: \mathbb {R} \rightarrow \mathbb {R}$ be differentiable on $\mathbb {R}$ and satisfy $\inf_{t \in \mathbb {R}}\left|f^{\prime }(t)\right| \geqslant q > 1$. Prove that the equation $f(t) = t$ has a unique solution on $\mathbb {R}$.

Prove that the equation $x(t) = t+\varepsilon x\left(t^{\alpha }\right)$, where $\left|\varepsilon \right| < 1, \alpha > 0$, has a unique solution in the space $C[0,1]$.

Prove that the mapping $A: C[0,1] \rightarrow C[0,1], A x(t) = \lambda \int_{0}^{t} x(s) d s+1$, is a contraction for $\left|\lambda \right| < 1$, and find its fixed point.

Let, in a complete metric space $X$, the mapping $f$ be continuous and let some power $f^{p} = \underbrace{f \circ f \circ \ldots \circ f}_{p}$ of the mapping $f$ be a contraction: $\rho \left(f^{p}(x), f^{p}(y)\right) \leqslant q \rho (x, y)$, where $q < 1, x, y \in X$. Prove that the mapping $f$ has a unique fixed point in $X$: $f(x) = x$.

Prove that any continuous mapping of a segment into itself has a fixed point.

Let $\bar{B}^{\prime }(0,1)$ be the punctured closed unit ball of the space $\mathbb {R}^{n}$ (i.e., the ball with the center removed). Give an example of a continuous mapping of this set into itself having no fixed point.

Does there exist an incomplete metric space in which the contraction mapping principle holds?

Construct an example of a complete metric space $X$ and a mapping $f: X \rightarrow X$ such that $\rho (f(x), f(y)) < \rho (x, y)$, but $f$ has no fixed point.