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I am facing the following definition of topology:

A topology on a set $X$ is defined as a subset $\cal O$ of the power set $\cal P$ of $X$, that:

  • contains the empty set and $X$
  • is stable over the arbitrary union
  • is stable over the finite intersection

I was explained the last one in the following way: taken a finite number of subsets of $\cal O$, their intersection belongs to $\cal O$. However, the intersection of an infinite number of subsets of $\cal O$ may not belong to $\cal O$.

Honestly, this does not make sense to me. It seems to me that, taken infinite subsets of a set, their intersection by definition will belong to the set, so that the topology is inherently stable under infinite intersection.

Would you please explain me the concept of stability under finite intersection?

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    $\begingroup$ Is the intersection of infinitely many open intervals an open set? $\endgroup$
    – William Elliot
    Oct 14, 2019 at 0:36
  • $\begingroup$ Let $\mathcal{O}$ be the topology generated by open intervals on $\mathbb{R}$. Take the intersection of finitely many open intervals $(a,b)$, for $a,b\in \mathbb{R}$ and you are guaranteed to get an open interval back, but take the infinite intersection over $(0, 1 + 1/n)$ for all $n\in \mathbb{N}$, and you get $(0,1]$, which is not open. You might be confusing the fact that every element in the intersection is definitionally contained within all the intersected sets with the question of whether this intersection set is an element of $\mathcal{O}$, which is a very different thing. $\endgroup$
    – Jack Crawford
    Oct 14, 2019 at 0:43
  • $\begingroup$ See math.stackexchange.com/q/284970. $\endgroup$
    – Paul Frost
    Oct 16, 2019 at 13:30

1 Answer 1

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The explanation you quoted, "taken a finite number of subsets of $\mathcal O\ \dots$" should have had "elements of $\mathcal O$" instead of "subsets of $\mathcal O$." The sets that you are intersecting are subsets of $X$, hence elements of the power set $\mathcal P$, and elements of $\mathcal O$. And the last of the three requirements in the definition of a topology says that the intersection of any finitely many elements of $\mathcal O$ is again an element of $\mathcal O$.

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