Why impose open sets in the defintion of a manifold? The definition of a topological manifold X specifies that the atlas must consist of open subsets of X.
It can be easily understood for a differential manifold since the transitions maps must have open subsets as domain and codomain in order to be able to talk about differentiability.
But why is so for a pure topological manifold ? Are there examples of a topological space endowed with an atlas whose domains are not open subsets, which does not fit into the intuitive idea of "gluing open balls together" ?
 A: I think you're misunderstanding something. An atlas is not a collection of open sets--it is a collection of local charts, ie homeomorphisms from open sets of your manifold to $\mathbb{R}^n.$ The reason you want these domains to be open is because open sets of your manifold are the ones which respect the topology.
Every topological-space would be a 0-dimensional manifold if you didn't require the sets to be open: Each singleton is homeomorphic to $\mathbb{R}^0$, that makes your atlas. But the problem is that sets carry topological data, and the subspace topology of a set loses a lot of information in general--it's only when taking the subspace topology on open sets that you retain this topological information.
For instance, if you don't require openness, then $\mathbb{R}^2$ is a 1-dimensional manifold: For each $t\in\mathbb{R},$ the line $x=t$ in the plane is homeomorphic to $\mathbb{R},$ and so you can use these lines and the obvious homeomorphisms as your atlas. But this doesn't respect the topology of $\mathbb{R}^2,$ as we're essentially treating it like a disjoint union of lines, when in $\mathbb{R}^2$ those lines 'see' each other.
