An error of Lebesgue. 1905 or so. Take a Borel set in the plane, project it onto a line, the result is a Borel set. Obvious: the projection of an open set is open, and the Borel sets in the plane are the least family containing the open sets, closed under countable unions and countable intersections.

But wrong. Projection doesn't commute with countable intersection.

Studying this error lead Suslin to begin the line of study now called "descriptive set theory", 1917 or so.