When is $\bold N$ Lindelöf?

summary:Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\Bbb N$ is a Lindelöf space, (2) $\Bbb Q$ is a Lindelöf space, (3) $\Bbb R$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $\Bbb R$ is separable, (6) in $\Bbb R$, a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$, (7) a function $f:\Bbb R\rightarrow \Bbb R$ is continuous at a point $x$ iff $f$ is sequentially continuous at $x$, (8) in $\Bbb R$, every unbounded set contains a countable, unbounded set, (9) the axiom of countable choice holds for subsets of $\Bbb R$