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:
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.
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 r<sub>i</sub> 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 a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub> in the plane R<sup>2</sup> and let be the dilation of ratio 1/2 around a<sub>i</sub>. 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.
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.