An atlas pulled back by a homeomorphism

For every topological model space $H$, every topological space $M$ with a specified charted-space structure modeled on $H$, every topological space $N$, and every homeomorphism $e:N\to M$, the construction defines a charted-space structure on $N$ modeled on the same $H$. Its atlas consists exactly of the charts obtained by first applying $e$ and then an atlas chart $c$ of $M$: such a chart has source $e^{-1}(\operatorname{source}(c))$, target $\operatorname{target}(c)$, forward map $c\circ e$, and inverse map $e^{-1}\circ c^{-1}$. The preferred chart at each $x\in N$ is obtained from the preferred chart of $M$ at $e(x)$ by this same composition. The topology on $N$ is its given topology, and no earlier chart structure on $N$, differentiability, manifold compatibility, dimension, separation, or nonemptiness hypothesis is required; empty spaces are included whenever the stated homeomorphism exists.