Infinite Intersections Of Open Sets

Geometry Level 4

Let $$U_{\alpha}$$ $$(\alpha \in A)$$ be a collection of open sets in $${\mathbb R}^2.$$ If $$A$$ is finite, then the intersection $$U = \bigcap\limits_\alpha U_{\alpha}$$ is also an open set. Here is a proof:

Suppose $$x\in U.$$ For each $$\alpha \in A,$$ let $$B_{\alpha}$$ be a ball of some positive radius around $$x$$ which is contained entirely inside $$U_{\alpha}.$$ Then the intersection of the $$B_{\alpha}$$ is a ball $$B$$ around $$x$$ which is contained entirely inside the intersection, so the intersection is open.

(Here a ball around x is a set $$B(x,r)$$ ($$r$$ a positive real number) consisting of all points $$y$$ such that $$|x-y|<r.$$ In $${\mathbb R}^2$$ it is an open disk centered at $$x$$ of radius $$r.$$)

Where does this proof go wrong when $$A$$ is infinite?

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