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We study the topological properties of attractors of iterated function systems (IFS) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation IFS: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, IFS (of first generation), or it possesses a particular dissection property, the attractor of the second generation IFS is the union of a finite number of closed intervals. We also prove a theorem that generalizes this result to certain infinite sums of compact sets, in the sense of Minkowski and under the Hausdorff metric. [ABSTRACT FROM AUTHOR]