On proofs of the $C^0$ general density theorem

We show that if $M$ is a compact manifold, then there is a residual subset ${\mathcal{N}}$ of the set of homeomorphisms on $M$ with the property that if $f\in {\mathcal{N}}$, then the periodic points of $f$ are dense in its chain recurrent set. This result was first announced in [4], but a flaw in that argument was noted in [1], where a different proof was given. It was recently noted in [5] that this new argument only serves to show that the density of periodic points in the chain recurrent set is generic in the closure of the set of diffeomorphisms. We show how to patch the original argument from [4] to prove the result.