Is the collection of atlases on a set $X$ a set?

Question

Well, the title says it all. I need to know if i can view the collection of all atlases on a given set $X$ as a ordinary set. Is this possible ? All the atlases are only topological atlases, no additional structure.

Thanks in advance!! Greetings...

Answer 1

If the charts are to $\mathbb{R}^k$ then it is a subset of $\mathcal{P}(\mathcal{P}(\mathbb{R}^k\times \mathcal{P}(X))$, by identifying a chart with a pair $(f \colon X \to \mathbb{R}^{k},U \subset X)$. If the dimension is not fixed a priori, then replace $\mathbb{R}^k$ with $\mathbb{R}^* = \bigsqcup_{k \ge 0} \mathbb{R}^k$.