Unclear use of big union symbol in proof of a fixed point theorem

Question

I'm looking at a proof given here: https://math.stackexchange.com/a/1885829/666263

Specifically, this section:

III. Lemma (Knaster–Tarski Fixed Point Theorem). If a function $\varphi:P(A)\to P(A)$ satisfies the condition $$X\subseteq Y\implies\varphi(X)\subseteq\varphi(Y)$$ for all $X,Y\subseteq A,$ then $\varphi(S)=S$ for some $S\subseteq A.$

Proof. Let $\mathcal F=\{X:X\subseteq\varphi(X)\}$ and let $S=\bigcup\mathcal F.$ First note that, if $X\in\mathcal F,$ then $X\subseteq S$ and $X\subseteq\varphi(X)\subseteq\varphi(S).$ Since every member of $\mathcal F$ is a subset of $\varphi(S),$ it follows that $S=\bigcup\mathcal F\subseteq\varphi(S),$ that is, $S\in\mathcal F.$ From $S\in\mathcal F$ it follows that $\varphi(S)\in\mathcal F,$ whence $\varphi(S)\subseteq S,$ and so $\varphi(S)=S.$

In the proof, the author writes $\mathcal F=\{X:X\subseteq\varphi(X)\}$ and then $S=\bigcup\mathcal F$. and i'm unclear whether $S = \bigcup_{A \in \mathcal{F}} A$ or something else. In other words, i'm asking if given a set of sets $\mathcal{F}, $ $\bigcup\mathcal F$ is the union of all elements of $\mathcal{F}$ or something else?

I've taken the liberty of adding the relevant portion of the proof you're asking about. Making your problem more self-contained will make answering easier: when potential answerers have to click through to other images/posts/links, it reduces the likelihood they'll spend time on your question. — KReiser, Jan 09 '20 at 05:14

KReiser · Answer 1 · 2020-01-09T08:05:55.747

2

$\mathcal{F}$ is a set consisting of subsets of $A$. The notation $S=\bigcup \mathcal{F}$ is intended to express the union of all of these subsets of $A$ as a subset of $A$.

edited Jan 09 '20 at 08:05

answered Jan 09 '20 at 05:16

KReiser

65,137

1

Actually that would not be a better way to write it. – Asaf Karagila Jan 09 '20 at 07:32
@AsafKaragila you're right, that isn't quite it. The original notation does not satisfy me, though - do you have a suggestion? – KReiser Jan 09 '20 at 08:08
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Getting used to the notation? :-) – Asaf Karagila Jan 09 '20 at 09:33
I knew there was a reason I leave the set stuff to folks like you :P – KReiser Jan 09 '20 at 09:45
I mean, why would you use $\sum_{i=0}^n a_i$ and not $a_0+\dots+a_n$? And why would you also end up writing $\sum a_i$ when the index set is clear from context? – Asaf Karagila Jan 09 '20 at 09:48
Well, the original issue in this case was that what was going on wasn't clear to the asker. I think saying "take the union of all the sets in $\mathcal{F}$" is probably the easiest way to clear this up if the notation isn't clear. – KReiser Jan 09 '20 at 09:58
I have no qualms with your answer as a whole, just that this is notation to get used. – Asaf Karagila Jan 09 '20 at 09:59

Unclear use of big union symbol in proof of a fixed point theorem

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