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I was reviewing topology and ZFC and noticed that the definition of a topology is defined in terms of collections and families of subsets:

A topology $T$ on $X$ is a collection of subsets of $X$, called open subsets, satisfying

  1. $X$ and $\emptyset$ are open
  2. The union of any family of open subsets is open
  3. The intersection of any finite family of open subsets is open

I was trying to define this definition formally in terms of ZFC, but I am not sure how to precisely formalize collections and families in ZFC, since they may not even fit inside of a set.

Does this mean ZFC on its own is not sufficient to describe topologies using the definition above?Also, where can I learn more about how things like collections and families are handled in mathematics?

Thanks!

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    $\begingroup$ In this context, "collection" and "family" are just synonyms of "set" in order to avoid confusion. For a lot of elementary readers, it is earlier to grok "a collection of sets" than "a set of sets". $\endgroup$
    – Xander Henderson
    Apr 15, 2021 at 0:45
  • $\begingroup$ You totally can. Just replace set collection and family with set. The only problem you might run into is if you're spaces are "too big" and have more than a set's worth of subsets, but that never actually happens in any meaningful situation. $\endgroup$
    – memerson
    Apr 15, 2021 at 0:46
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    $\begingroup$ @memerson: Never is a little too strong: one can and does define a topology on $\mathbf{No}$, the class of surreal numbers. $\endgroup$ Apr 15, 2021 at 0:55
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    $\begingroup$ @BrianM.Scott Or the various spaces-of-models associated to "large" logics (with a set-sized logic the $T_0$ification is guaranteed to be set-sized, but not so with a logic like $\mathcal{L}_{\infty,\omega}$). $\endgroup$ Apr 15, 2021 at 1:43

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There's no issue here. Terms like "collection" are used here instead of "set" for purely pedagogical purposes (it helps the reader keep track of the "type" of object under consideration). More plainly in the language of $\mathsf{ZFC}$, we have:

A topological space is a pair $(X,\tau)$ where $\tau\subseteq\mathcal{P}(X)$ such that $\emptyset\in\tau$, $X\in\tau$, $\forall a\in\tau,b\in\tau(a\cap b\in\tau)$, and $\forall c\subseteq\tau(\bigcup c\in\tau)$.

I've used abbreviations in the above definition, e.g. "pair," "$\mathcal{P}(-)$," "$\subseteq$," etc., but these are all straightforward to unpack in terms of $\in$.

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  • $\begingroup$ oh, of course. Because the topology is defined in terms of subsets it must be able to fit inside a set. The power set notation makes it very clear. Thanks! $\endgroup$
    – David Hill
    Apr 15, 2021 at 2:28

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