G-delta and F-sigma sets

Prove that each open subset of a metric space $V$ is the union of some sequence of closed subsets of $V$.

Prove that the set of irrational numbers is a $G_{\delta }$ set.

Prove that in a metric space every closed set is a $G_{\delta }$ set and that every open set is an $F_{\sigma }$ set.

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. Conclude that there is no real-valued function on $\mathbb {R}$ that is continuous at the rational numbers only.

Let $E$ be a subset of a complete metric space $M$ and suppose that both $E$ and $E^{C}$ are dense. Prove that at most one of them is the union of a countable collection of closed sets. Conclude that the set of rational numbers in $[0,1]$ is not the intersection of a countable collection of open sets.

Prove that the set of rational numbers is not a $G_{\delta }$ set.

Prove that in a complete metric space, a set is of the first category if and only if its complement contains a dense $G_{\delta }$ set.

Let ( $M, d$ ) be a metric space and let $G$ be a $G_{\delta }$ subset of $M$. Prove that $G$ is of the second category in itself.

Show that each open set is an $F_{\sigma }$ set.

Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions defined on $\mathbb {R}$. Show that the set of points $x$ at which the sequence $\left\{ f_{n}(x)\right\}$ converges to a real number is the intersection of a countable collection of $F_{\sigma }$ sets.

A set of real numbers is said to be a $G_{\delta }$ set provided that it is the intersection of a countable collection of open sets. Show that for any bounded set $E$, there is a $G_{\delta }$ set $G$ for which

Show that

Let $X$ be a topological space, $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ a $\sigma$-finite Radon measure, and $E \subseteq X$ a Borel set. Show that there is a $G_{\delta }$ subset $A$ of $X$ and an $F_{\sigma }$ subset $B$ of $X$ for which

Let $X$ be a separable, compact, Hausdorff space. Show that every closed set is a $G_{\delta }$ set.