2:36 AM
I was a bit unsure about tagging this question.
2

Is there a topology induced in $\mathcal P (X)$ for an infinite set $X$? Intuitively, if $s_1 \supseteq s_2 \supseteq s_3 ...$ and $\bigcap_i s_i = s$ then we could say $s$ is the limit of $\{s_i\}$. But I don't exactly know how to define the open sets to make that formal. Even better: Is there...

However, I have decided to add and . My reasoning was approximately like this:
There is a notion of $\limsup X_n$ and $\liminf X_n$ of sets. And if we follow the tag-wiki questions about them we should use .
We can also say that $\lim X_n=X$ if $\limsup X_n=\liminf X_n=X$.
If I am not mistaken, the convergence of decreasing system of sets described in the question is special cases of this. So I guess it should be tagged the same.
The tag-info for limsup-and-liminf says: "For questions concerning the limsup/inf of sets, please add the tag. For questions involving abstract partial orders, use also the tag."

1 hour later…
3:43 AM
@AsafKaragila Did you have time to think about your suggestion for ? Or is it better to keep and separately? Should we bring this up on meta?
Or perhaps will we wait for 2017 tag management thread? Coming soon to your meta.

4:01 AM
I have added and here. Do they fit? Any other suggestions?
13

$$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$ You can always break up infinity/infinity into the left hand side, where n is an arbitrary number. However, on the left hand side $\frac{n}{\infty}$ is always equal to 0. Thus $\frac{\infty}{\infty}$ should always equal $0$.