Intersection of infinite set of open sets

[maiden quest]

take the infinite set of intervals (a-1/n,a+1/n).
Say the intersection of all of them are open, then {a} is open for all a in R, however then every subset of R is open

so openness loses meaning with infinite intersections

as everything is open

(also, topologies are defined by finite intersections of open sets being open)

also for infinite S's, your choice of epsilon becomes an infimum, so there is possibility you get eps=0

your choice of epsilon need not be positive with infinite S

like take the intervals I wrote, then eps = inf{1,1/2,1/3,...}=0