Abstract:\par \noindent {Theorem.} {In ZF {(}i.e., Zermelo-Fraenkel set theory without the axiom of choice{)} the following conditions are equivalent: \roster \item "(1)" $\Bbb N$ is a Lindel\"of space, \item "(2)" $\Bbb Q$ is a Lindel\"of space, \item "(3)" $\Bbb R$ is a Lindel\"of space, \item "(4)" every topological space with a countable base is a Lindel\"of space, \item "(5)" every subspace of $ \Bbb R$ is separable, \item "(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 $x$, \item "(7)" a function $f:\Bbb R\rightarrow \Bbb R$ is continuous at a point $x$ iff $ f$ is sequentially continuous at $x$, \item "(8)" in $\Bbb R$, every unbounded set contains a countable, unbounded set, \item "(9)" the axiom of countable choice holds for subsets of $ \Bbb R$. \endroster }
Keywords: axiom of choice, axiom of countable choice, Lindel\"of space, separable space, (sequential) continuity, (Dedekind-) finiteness
AMS Subject Classification: Primary 03E25, 04A25, 54D20; Secondary 26A03, 26A15, 54A35