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In Munkres, the definition of topology is given as the following:

Definition. A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ having the following properties:

  1. $\varnothing$ and $X$ are in $\mathcal{T}$.
  2. The union of the elements of any subcollection of $\mathcal{T}$ is in $\mathcal{T}$.
  3. The intersection of the elements of any finite subcollection of $\mathcal{T}$ is in $ \mathcal{T}$.

A set $X$ for which a topology $\mathcal{T}$ has been specified is called a topological space.

In the third condition, when we are considering the finite subcollection, we are kind of allowing the subcollection to be the empty set, (he is allowing the finite set to be empty) but the intersection of the elements of empty collection is not defined in his book. (He said it is reasonable to say that if one has a given large set X that is specified at the outset of the discussion to be one's "universe of discourse," then we can say $\cap \varnothing = X$, but to avoid difficulty, we shall not defined the intersection when the collection is empty) I feel like there is some ambiguity here. I want a self-contained definition within his book, am I understanding something wrong here? Any help is appreciated!

Zhigang Wu
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2 Answers2

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Good question. The convention (which Munkres mentions) is that the intersection of no sets is the whole space $X$. That is in $\mathcal{T}$.

This post When indexing set is empty, how come the union of an indexed family of subsets of S is an empty set? explains why the convention is reasonable.

But even without it, Munkres's definition of a topology is rigorous since he explicitly addresses the issue and says he will avoid it.

Ethan Bolker
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  • How is Munkres's definition rigorous if first he leaves the intersection of the empty collection undefined, then requires the intersection of the empty subcollection to be in $\mathcal{T}$? – Adayah Jun 20 '25 at 08:05
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    @Adayah There's no problem with undefined things you never refer to. This is a rigorous definition for his purposes because he explicitly says he will never need an empty intersection in theorems he proves. He does not require the empty intersection to be the whole space, he just tells the reader some people do. – Ethan Bolker Jun 20 '25 at 11:19
  • But he does refer to the empty intersection precisely in condition 3 that the question is about. Unless he wrote in advance something to the effect of "whenever the evaluation of a condition requires us to compute the intersection of the empty family, we implicitly exclude that family from the range of the relevant quantifier"...? – Adayah Jun 21 '25 at 18:28
  • @Adayah I think the disclaimer here when he defines a topology is sufficient. To show me wrong you'd have to search the book to find a place where there is an intersection of a family that might be empty. – Ethan Bolker Jun 21 '25 at 21:14
  • Once again: condition 3, which this whole topic is about, speaks of intersection of a family that might be empty. No offense, but how many times will I have to repeat that? – Adayah Jun 22 '25 at 16:33
  • @Adayah We just have to agree to disagree. – Ethan Bolker Jun 22 '25 at 19:12
  • @Adayah I think the key here is that whether the statement "The intersection of the elements of any finite subcollection of T is in T" is true given that the empty intersection is not defined. I think it is true because if you take any finite subcollection, say T', of T (since you have already taken it, it has to be something that is DEFINED), therefore, it is in T. – Zhigang Wu Jun 23 '25 at 18:26
  • @ZhigangWu I think the statement is not only not true, but not even well defined. The fixed finite subcollection $T'$ of $T$ is defined, but if it happens to be the empty subcollection, its intersection is not defined. Then, programmatically speaking, the computation of the condition's truth value returns an error. – Adayah Jun 24 '25 at 15:48
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The question Does a family of subsets have to be covering for it to be the subbase of the topology it generates? has a very similar background: What is the empty intersection?

Munkres avoids this discussion by stating on p.13

To avoid difficulty, we shall not define the intersection when $\mathcal A$ is empty.

This means in particular that in 3. the empty subcollection of $\mathcal T$ is not considered. Yes, the empty subcollection is finite, but by Munkres's convention it is not considered in 3. You are right, this is a bit unsatisfactory and he should better have written

  1. The intersection of the elements of any non-empty finite subcollection of $\mathcal T$ is in $\mathcal T$.

or

  1. The intersection of the elements of any finite subcollection of $\mathcal T$, if it is defined, is in $\mathcal T$.

Anyway, as Munkres points out, the only reasonable way to define the intersection of the empty family of subsets of $X$ is $$\bigcap \emptyset = X .$$ See p.12. However, that $X \in \mathcal T$ is covered by 1. which shows once more that considering $\bigcap \emptyset$ is unnecessary.

Similarly, the empty union is $\emptyset$ (which does not cause any problems). Thus 2. implies $\emptyset \in \mathcal T$ - and this is again covered by 1.

Many authors do not use 3. but require

If $U, U' \in \mathcal T$, then $U \cap U' \in \mathcal T$.

By induction this implies that the intersection of the elements of any non-empty finite subcollection of $\mathcal T$ is in $\mathcal T$.

You see that the spirit of the definition of a topology is to avoid considering empty subcollections of $\mathcal T$; property 1. makes this superfluous.

Paul Frost
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