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I'm learning topology and I'm reading Dugundji's General Topology book. He gives the definition of axioms of a topology $\tau$ on $X$:

  • 1) & 2) (a topology is closed under arbitrary union and finite intersection )
  • 3) $\emptyset$ and $X$ belong to topology $\tau$.

then he says: "*Observe that since the union (resp. intersection) of an empty family of sets in X is $\emptyset$ (resp. X) axiom 3 is actually redundant. *"

My question are,

why he consider a empty family of sets of $X$ in the observation?

whitout axiom 3, how do we know that in fact the empty set belongs to the topology $\tau$?

I really don't understand his observation. Any help would be appreciated.

HeMan
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    He consider that because you can do that. That is: You could want to do that. He is speaking about any possible unions and intersections, and this (empty colection) is one of them. – Red Banana Jan 21 '17 at 21:46
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    Because vacuously an empty union (the union of zero sets) is empty. So the empty set is the union of zero sets of X. So by axiom 1 it is in the topology. Likewise X is the intersection of zero sets of X. So by axiom 2 it is in the topology. – fleablood Jan 21 '17 at 22:19

1 Answers1

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Property 1 says an arbitrary union of open sets is open. An arbitrary union of sets $$\bigcup_{A\in \mathcal{A}} A$$ is the union of all the sets in some collection $\mathcal{A}$ of sets. In particular, you could take $\mathcal{A}=\emptyset,$ the empty collection of sets. The union over the empty collection is the empty set (what else?). Thus property 1 implies that the empty set is open.

Likewise the intersection over the empty collection is naturally defined as $X.$ This makes sense since intersecting with more and more sets restrict the elements more and more. It makes sense to 'start out' with all the elements, so the intersection of the empty collection of sets is $X.$ Also, it is a finite intersection (since the empty collection is finite). Thus, property 2 implies that $X$ is open.

EDIT: As commenters have emphasized, at the end of the day, assigning a value to an intersection of an empty collection of sets is a convention. It only makes sense (to my knowledge) if you have a notion of a total space $X$ that every set under consideration is a subset of, as is the case in topology. In this scenario, the notion is well-defined and I tried to make it intuitive in my answer.

Formally, in the scenario with a total space $X,$ the definition of the union and interection are given by $$\bigcup_{A\in \mathcal{A}} A = \{x\in X\mid \exists A\in\mathcal{A}\mbox{ such that } x\in A \}\\\bigcap_{A\in \mathcal{A}} A = \{x\in X\mid \forall A\in\mathcal{A},\; x\in A \}.$$ Applying the definition for the intersecton in the case $\mathcal{A}=\emptyset$ gives the result $X$ as indicated when you apply the notion of vacuous truth to the quantified statement $\forall A\in \emptyset,\ldots$, in which case it is always true regardless of what $\ldots$ is.

So while it's true that the 3rd property is redundant, this fact is somewhat formal and not all that interesting. Thus why pretty much everyone retains the 3rd property for clarity.

spaceisdarkgreen
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  • To the OP, in particular, that the empty intersection is naturally defined as $X$ is a consequence of vacuous truth https://en.wikipedia.org/wiki/Vacuous_truth, something which some people (myself included) might find unintuitive when they encounter it for the first time. Also, the cardinality of the empty set is $0$ (written: $|\emptyset|=0$), so empty intersection counts as a finite intersection. – Chill2Macht Jan 21 '17 at 22:08
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    The fact that vacuous truths are unintuitive, and even if you understand them, feel like bootstrapping and cheating, and should be exceptions, is precisely why we add them as a 3rd axiom. – fleablood Jan 21 '17 at 22:22
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    That the union of an empty family of sets is empty is not a convention. – Mariano Suárez-Álvarez Jan 22 '17 at 03:14
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    In axiomatic set theory, the intersection of a collection of sets is defined only for a non-empty collection The modified definition of intersection you use here works, but it's a (yes, natural) convention which needs to explicitly stated. If it isn't, then $X\in \tau$ doesn't follow from the standard definition of intersection, and it should be stated as a 3rd condition of the definition. See my answer here – user019828 Jul 13 '21 at 22:04