Let U be any open set. Then, every point y in U is in the center of an open ball B_y that is contained in U.

Thus, every point y of U is in the union of all those open balls B_y.Further, since each of those balls is contained in U, so is their union.

The last two statements show that U is in fact the union of the open balls B_y;

U = _y B_y.

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