Fixed point free homeomorphisms of the complex plane

Our purpose in this article is to prove that the group $H(\mathbf{C})$ of homeomorphisms of the complex plane $\mathbf{C}$ is a metric group equipped with the metric induced by uniform convergence of homeomorphisms and their inverses on compacts and the set $$\{h\in H(\mathbf{C}):(\mathrm{\forall }z\in \mathbf{C})(h(z)\mathrm{\ne }z)\}$$ of fixed point free homeomorphisms of the complex plane is a conjugacy invariant dense $G_{\delta }$ subset of $H(\mathbf{C})$.