I am just curious about the axioms of topology. In particular with regard to finite intersections. The way that I imagine the axioms of topology is that we give a set $X$ a way of arranging it's points. I was wondering why do we restrict ourself to only finite intersections. Does someone has any intuition about this?
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$\begingroup$ An infinite intersection can "collapse" onto a "closed" limit point. If $A_n = (-\frac 1n, \frac 1n)$ then $\cap_{i\in\mathbb N} A_i = \{0\}$ not an open set. Or $B_n =(-\frac 1n, 1 +\frac 1n)$ the $\cap_{i\in\mathbb N} B_n = [0,1]$. $\endgroup$– fleabloodAug 5, 2019 at 22:30
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$\begingroup$ Yeah, that is what I was guessing. Thanks for clarifying and also pining that down! $\endgroup$– user693839Aug 5, 2019 at 22:33
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$\begingroup$ See also math.stackexchange.com/q/2884822. $\endgroup$– Paul FrostAug 5, 2019 at 22:40
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