Infinite Intersections Of Open Sets

Geometry Level 4

Let Uα U_{\alpha} (αA) (\alpha \in A) be a collection of open sets in R2. {\mathbb R}^2. If AA is finite, then the intersection U=αUα U = \bigcap\limits_\alpha U_{\alpha} is also an open set. Here is a proof:

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

((Here a ball around xx is a set B(x,r) B(x,r) (rr a positive real number) consisting of all points y y such that xy<r.|x-y|<r. In R2 {\mathbb R}^2 it is an open disk centered at xx of radius r.)r.)

Where does this proof go wrong when AA is infinite?

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