In Baire spaces, is the following always true: the intersection of a countable sequence of decreasing open sets is non-empty?
No. Consider the open intervals (0,1/n) on the real line.
Answer is NOT. Consider R with natural topology which is Baire space. Get the countable sequence U_{n}=(0; 1/n). The intersection of all these sets is empty.
The link contains some information concerning intersections of open sets.
http://math.stackexchange.com/questions/740020/questions-related-to-intersections-of-open-sets-and-baire-spaces
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