Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\mathbb{N}$ is a Lindelöf space, (2) $\mathbb{Q}$ is a Lindelöf space, (3) $ℝ$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $ℝ$ is separable, (6) in $ℝ$, 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 $f:ℝ\to ℝ$ is continuous at a point $x$ iff $f$ is sequentially continuous at $x$, (8)...