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I'm reading a lecture note in topology. There is a remark

Remark 1.2. A quick induction shows that any finite intersection $U_{1} \cap \cdots \cap U_{k}$ of open sets is open. It is important to point out that it is in general not true that an arbitrary (infinite) union of open sets would be open, and it is often difficult to decide whether it is so.

I think the part about an arbitrary (infinite) union of open sets is wrong. Do I miss something here?

Akira
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    By definition, any union of open sets is open. It's for intersections that things could go very false. I think that's what the text intended. – Henno Brandsma Oct 03 '21 at 09:57

2 Answers2

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No you don't miss anything here. It is most likely a typo and they mean that an arbitrary infinite intersection of open sets need not be open. For instance, consider $$\{0\}= \bigcap_{n=1}^\infty (-n^{-1},n^{-1}).$$

J. De Ro
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There's a typo by referring to 'union'. As for intersections, it's not wrong: by induction, you can show that for every $n$, the intersection of $n$ open sets is open, but this is not possible for a non-finite amount of sets. This is consistent with the fact that the 'usual' open sets in $\mathbb R^n$ (with the Euclidean metric) follow this behaviour: for example, $\bigcap\limits_{n=1}^\infty B(0,\frac{1}{n}) = \{0\}$ which isn't open.

Miguel González
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