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I don't look for exactly philosophical/motivational reason, i'm just confused because the almighty wikipedia says that $\forall A: \varnothing \subseteq A$, hence requiring empty set seems redundant (especially for this fundamental branch of math). Unless topology is not a set. But the definitions that i saw say that topology is a subset of the power set.

i also read this but they apparently discussing a bit of a different problem.

fyi my understanding of mathematics is calc-2 and everything around that

Utof
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    The axioms for a topological space $(X, \tau)$ require $\emptyset \in \tau$, not $\emptyset \subseteq \tau$. You are right that the second condition is automatic, but only the first one is aa axiom for topologies – Tzimmo Nov 04 '23 at 14:45
  • At a minimum it makes definitions cleaner and avoids special cases in proofs. For example, under the definition that a set is closed if its complement is open, it follows that the entire space is closed. – AlgTop1854 Nov 04 '23 at 14:47
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    A topology on a set $X$ is a family of subsets of $X$, you can think about it as if you are "highlighting" certain subsets of interest in your space. It is then by definition that we require a topology to always "highlight" the empty set, and similarly for the entire space, the fact that every set contains the empty set tells us very little in this case, as you could easily have an open set (i.e. a subset in your topology) which contains a subset which is not in your topology, take as an example $\mathbb{R}$ with the usual definition for open sets and consider a closed interval in an open one – Spasoje Durovic Nov 04 '23 at 14:55
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    Actually, requiring $\varnothing\in\tau$ is indeed redundant, but for a more subtle reason than your probable confusion between $\in$ and $\subseteq$ expained by @Tzimmo's comment: $\varnothing\in\tau$ because $\tau$ is already required to be closed under unions and $\varnothing$ is the union of the empty family of sets. – Anne Bauval Nov 04 '23 at 15:04
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    @AnneBauval You're right, I missed that. – Tzimmo Nov 04 '23 at 15:28
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    In a topological space, we have the nice fact: "the intersection of two open sets is open". We would not want to have to change this to: "the intersection of two open sets is either open or empty". – GEdgar Nov 04 '23 at 15:38

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The axioms for a topological space $(X, \tau)$ require $\emptyset \in \tau$, that is, the topology contains the empty set as an element. You remarked that $\emptyset \subseteq \tau$ is automatic, that is, the topology always contains the empty set as a subset.

These notions of elements and subsets are different: Let $A = \{13, \{42\}\}$. Then $\{13\} \subseteq A$, but $\{13\} \notin A$. On the other hand, $\{42\} \in A$, but $\{42\} \nsubseteq A$ since $42 \notin A$.

In the comments, @AnneBauval remarked that even the axiom $\emptyset \in \tau$ is redundant. In a topological space, arbitrary unions of open sets are open. In particular, this is true for the union of the empty family (that is, take the union of no sets at all). This union is the empty set.

Tzimmo
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    ah, i see, and thanks for the example, made it all much clearer! I wonder if there are intuitive examples of the ramifications of this slight difference (not a singleton subset, but a singleton as an element of a set)

    and to clarify:

    1. is there a precise notion of a "family"?
    2. We could take the union, we could also take the intersection, getting the empty set? or in some examples it could not result in empty set?
    3. this is my first instance of discovering something redundant in math definition, woah. So the definition will probably stay like that because of the historic reasons?
     – Utof Nov 05 '23 at 16:25
  • Technically, a family of elements of the set $A$ is just a map $I \to A, i \mapsto a_i$ for some set $I$. The main focus here is on the elements $a_i \in A$. The set $I$ is mainly used to access them. The empty family of open sets here is the map $\emptyset \to \tau$.
  • The intersection over the empty family would be considered to be the entire ambient space $X$. You can actually combine the axioms $X \in \tau$ and $U,V \in \tau \Rightarrow U \cap V \in \tau$ into "Finite intersections of open sets are open".
  • The redundant definition is much clearer. One might overlook empty families.
    • – Tzimmo Nov 05 '23 at 17:01