Can ZFC define topologies

general-topology

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!

Best Answer

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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