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To make the title a little bit more proecise:

Let $X$ be a set and a map, $F:\mathcal{P} (X) \rightarrow \mathcal{P} (X)$ (where $\mathcal{P} (X)$ denotes the powerset of $X$), such that $A \subseteq F(A)$ and $F(A) \subseteq F(B)$ for all $\mathcal{P} (X) \ni A \subseteq B \in \mathcal{P} (X)$.

If we call a set $T \in \mathcal{P} (X)$ closed if $F(T)=T$.

How can I prove that given a function $F$ like above, a map $F':\mathcal{P} (X) \rightarrow \mathcal{P} (X)$ exists, such that all sets that are closed under $F$ are also closed under $F'$ and such that $F'$ satisfies the same properties as $F$ and additionally the property $F'(F'(A))=F'(A)$ for all $A \in \mathcal{P} (X)$ ?

Somehow $F$ would have to be iterated infinitely many times, to obtain the idempotency, but I can't figure out, how to do that.

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Are you sure you are not missing some properties of $F$ in your question? Since with the above definition (monotone and increasing) it is possible that the only closed set is $X$. At least $F(\emptyset)=\emptyset$ seems to be rasonable in most situations. The reason why I am asking is that the usual definition of preclosure (or Čech closure) operator is slightly different. See and – Martin Sleziak Apr 24 '11 at 19:25

1 Answer 1

up vote 6 down vote accepted

One way is to do what you're thinking, that is, to try and iterate $F$ infinitely until it stabilizes, and then you'll have idempotency. This might require a transfinite iteration, so I'll explain this approach after.

A simpler approach is to let $F'(A)$ be the intersection of all the $F$-closed sets containing $A$. It's easy to check that this works. This is similar to the situation where you define a closed set in a topological space to be one whose complement is open, and then define the closure of a set $A$ to be the intersection of all closed sets containing $A$.

Now, for the iterative approach, we're going to define a transfinite sequence of functions $F_{\alpha} : \mathcal{P}(X) \to \mathcal{P}(X)$. Define $F_0 = F$, and $F_{\alpha + 1} = F\circ F_{\alpha}$. For limit ordinals $\delta$, define $F_{\delta}(A) = \bigcup _{\alpha < \delta} F_{\alpha}(A)$. For each $A$, we get an increasing transfinite sequence of subsets of $X$: $$A \subseteq F_0(A) \subseteq F_1(A) \subseteq \dots \subseteq F_{\alpha}(A) \subseteq \dots$$ This sequence must stabilize at some point, since $X$ is a set and the ordinals form a proper class. So for a given $A$, we set $F'(A)$ to be the eventual value of the transfinite sequence above.

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