Decreasing sequence of closed sets?

Question

Let $(M,d)$ be a compact metric space. Suppose that $(F_n)$ is a decreasing sequence of nonempty closed sets in $M$, and that $\bigcap_{n=1}^\infty F_n$ is contained in some open set $G$. Then $F_n \subset G$ for all but finitely many $n$.

I know that $\bigcap_{n=1}^\infty F_n \neq \emptyset$, but I'm having trouble proving the above statement.

A detailed answer to the question: https://math.stackexchange.com/questions/1840573/intersection-of-compact-sets-contained-in-open-set — hazelnut_116, Mar 27 '24 at 09:06

score 4 · Answer 1 · answered Nov 02 '15 at 07:45

4

Suppose not. Then there is an infinite sequence of $F_n$ such that all the $F_n$ aren't contained inside $G$. Since $G$ is open, each $F_n \setminus G$ is also closed; each is also nonempty by assumption. Consider the intersection of those.

answered Nov 02 '15 at 07:45

Patrick Stevens

36,135

I know the intersection of those $F_n \setminus G$ is closed, but I'm not sure how to proceed. – user286129 Nov 02 '15 at 07:53
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Is it true that $\bigcap_{n=1}^\infty (F_n\setminus G) = (\bigcap_{n=1}^\infty F_n) \setminus G$? – user286129 Nov 02 '15 at 08:20
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@user286129 . YES...Let $ G^c$ be the complement of $G$.Both sides are equal to $\cap_n(F_n\cap G^c)$. – DanielWainfleet Nov 02 '15 at 08:32
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Note that the closed sets are compact, nested intersections of non-empty compact sets are non-empy.. – Henno Brandsma Nov 02 '15 at 08:46

Decreasing sequence of closed sets?

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