The way limits are defined in Set Theory (lim sup = lim inf or limits of indicator functions) looks at odds compared to general definitions in term of topology, filters or ultrafilters. Ref: https://en.wikipedia.org/wiki/Set-theoretic_limit . Is there a way to express it similarly to general topological limits? Or the other way around (limits of sequence of real numbers considered as Dedekind cuts actually coincide with Set-theoretic limits)?
2026-04-07 01:06:39.1775523999
Is there a way to express the set-theoretic limit in terms of topology, filters or ultrafilters?
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Yes, if you put the right topology of $\mathcal P(X)$, the limit of a sequence of subset of $X$ is a topological limit.
The required topology can be described as the product topology on $\{0,1\}^{X}$, transferred to $\mathcal P(X)$ in the obvious way.
Unfolding one characterization of product topology, we can also describe this topology on $\mathcal P(X)$ as the topology generated by the sets $$ \{ A \cup C \mid C\subseteq (X\setminus B) \} $$ for all finite $A,B\subseteq X$.