Everywhere dense sets. Baire's theorem

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.