Open set condition
In fractal geometry, the open set condition (OSC) is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction. Specifically, given an iterated function system of  contractive mappings , the open set condition requires that there exists a nonempty, open set V satisfying two conditions: 

 The sets  are pairwise disjoint. Introduced in 1946 by P.A.P Moran, the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure. An equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the set is greater than zero. Computing Hausdorff dimension
When the open set condition holds and each  is a similitude (that is, a composition of an isometry and a dilation around some point), then the unique fixed point of  is a set whose Hausdorff dimension is the unique solution for s of the following:

where ri is the magnitude of the dilation of the similitude. With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three non-collinear points a1, a2, a3 in the plane R2 and let  be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping  is a Sierpinski gasket, and the dimension s is the unique solution of

Taking natural logarithms of both sides of the above equation, we can solve for s, that is: s = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. Strong open set condition
The strong open set condition (SOSC) is an extension of the open set condition. A fractal F satisfies the SOSC if, in addition to satisfying the OSC, the intersection between F and the open set V is nonempty. The two conditions are equivalent for self-similar and self-conformal sets, but not for certain classes of other sets, such as function systems with infinite mappings and in non-euclidean metric spaces. In these cases, SOCS is indeed a stronger condition. See also
Cantor set
List of fractals by Hausdorff dimension
Minkowski–Bouligand dimension
Packing dimension

References

Fractals
Iterated function system fractals