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Let $X$ be a closed convergence compact set. Let $A_1\subset ...\subset A_n\subset ...$ be a sequence of increasing closed subsets in the set $X$.

What other conditions do we need to impose of $X$ in order to ensure that the sequence converges?

"Sequence converge" means: there exists $A$ such that $A_n\to A$ in the closed convergence.

Motivation: this is closely related to the monotone convergence theorem of sets. For example, if we let the space be measurable, then it is well-known that the sequence converge. For another example, complete metric space also has a MCT.

I want to know what is the minimal conditions needed for the MCT to hold for sets in this abstract setting.

dodo
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    It depends on how you define convergence of sets – Hyperbolic PDE friend May 16 '24 at 08:47
  • @HyperbolicPDEfriend One sec – dodo May 16 '24 at 08:57
  • @HyperbolicPDEfriend "Sequence converge" means: there exists $A$ such that $A_n\to A$ in the closed convergence. – dodo May 16 '24 at 09:18
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    What's closed convergence? – Sassatelli Giulio May 16 '24 at 09:23
  • I echo Sassatelli Giulio, Searching for "closed convergence" offers no significant hits. So even if this concept is defined in some context, it is still not something you should expect others to know without explaining. In terms of set-theoretic convergence, it will always converge to $\bigcup_n A_n$ by the monotonicity. – Paul Sinclair May 17 '24 at 18:27
  • @PaulSinclair Let me double check. Thanks for the response. I wonder what is a more common appoach? – dodo May 24 '24 at 15:00
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    There are many types of convergence of sets that can be defined, depending on what structures are defined on the space. Set-theoretic convergence requires nothing beyond them being sets. The limit of $A_n$ converges if and only if every point that is in infinitely many of the $A_n$ is in all of the $A_n$ for sufficiently large $n$. With the limit being the set of all such points, Topologies and measures in particular allow for other convergence concepts. I do not know which "closed convergence" would mean. – Paul Sinclair May 26 '24 at 02:59

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