Let's say we were to define some generalizations of the notion of an algebraic closure for a field $F$ that is equipped with a metric $d$ (or something that behaves like a metric for the purposes of discussing radii of convergence). Some specific examples could be:
-a "power series closure", defined as a field extension in which every non-constant power series with coefficients in the closure has a root, and every element is a root of a power series with coefficients in $F$
-an "$r$-power series closure" as the closure for power series of convergence radius less than or equal to $r$
-series with some generalizations of polynomials and perhaps even uncountable indexing sets for the sums, whenever they can be defined
-generally any closure for a specific class of functions, such as continuous, differentiable, or analytic functions
Has this notion already been explored somewhere, and if so under what name(s)? If not, then I have several exploratory questions:
-When would such closures exist?
-When would they be unique up to isomorpism?
-If they are not unique up to isomorphism, would there be some interesting ways to classify them?
-What would their relationships be, beyond the obvious relationships for radii of convergence?
-What would they look like for $\mathbb{R}$?
I appreciate any input, thanks.