Open Sets, Closed Sets, and Borel Sets of Real Numbers

Is the set of rational numbers open or closed?

What are the sets of real numbers that are both open and closed?

Find two sets $A$ and $B$ such that $A \cap B = \emptyset$ and $\bar{A} \cap \bar{B} \neq \emptyset$.

A point $x$ is called an accumulation point of a set $E$ provided it is a point of closure of $E \sim \left\{ x\right\}$.

A point $x$ is called an isolated point of a set $E$ provided there is an $r > 0$ for which $(x-r, x+r) \cap E = \left\{ x\right\}$. Show that if a set $E$ consists of isolated points, then it is countable.

A point $x$ is called an interior point of a set $E$ if there is an $r > 0$ such that the open interval ($x-r, x+r$) is contained in $E$. The set of interior points of $E$ is called the interior of $E$ denoted by int $E$. Show that

Show that the Nested Set Theorem is false if $F_{1}$ is unbounded.

Show that the assertion of the Heine-Borel Theorem is equivalent to the Completeness Axiom for the real numbers. Show that the assertion of the Nested Set Theorem is equivalent to the Completeness Axiom for the real numbers.

Show that the collection of Borel sets is the smallest $\sigma$-algebra that contains the closed sets.

Show that the collection of Borel sets is the smallest $\sigma$-algebra that contains intervals of the form $[a, b)$, where $a < b$.

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