In subjects like Differential Geometry/ General Topology one often constructs for each $x$ in a space $X$ a set $U_x$ satisfying certain properties. Examples where one does constructions like this:
- Deducing that every open cover of a second countable space has a countable subcover.
- Proving that any covering space of a topological $n$ - manifold is second countable.
I am interested in whether there are any set theoretic issues like say use of the axiom of choice. In particular, how can we say "for each $x$ construct a set $U_x$.." simultaneously for all $x \in X$?
Also, my set theory is not so deep but I know that assuming that the set of all sets is a set leads to a contradiction in the style of Russell's paradox. However somehow it still makes sense to speak of a "maximal smooth atlas containing a given atlas" of a smooth manifold, or all closed sets containing a given set in a topological space.
Why is there no problem in doing this and we don't get to a situation like Russell's paradox?