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For example, if I want to write a closure of a set, which is denoted A_1, should I use $\overline{A_1}$ or $\overline A_1$ (or something completely different)?

For me, the more logical choice seems to be $\overline{A_1}$ (since I read \overline command as "closure of"). But maybe from typographical point of view this can be considered differently.

Here is a MathB.in link for comparison. (I am aware that rendering in MathJax and LaTeX is not necessarily the same, but I do not know what is standard way to make images with symbols for TeX.SE posts.)

Here is also a picture which I tried to create from pdf output using the method described in this answer on meta.

$\overline{A_1}$ vs. $\overline A_1$

I have seen that there exists a question \overline and \widehat with subscript. But that question asks about subscript added to the bar symbol, not subscript added to the symbol(s) which are under the bar.

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Certainly something like $\overline{A_1}$, because it is closure of $A_1$. But I understand, that the rule may look a bit too long for you. Indeed, there is no italic correction for math font. If you prefer a bit shorter version, see, e.g. answers to Appearance issues with \bar{} and \underline{}.

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You could think of ${\overline A}_1$ as being defined as $\overline{A_1}$ in which case you have the shorter line and can just write \overline A_1.

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