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{{Short description|Condition for self-similar fractals}}
[[File:Open set condition.png|thumb|an open set covering of the sierpinski triangle along with one of its mappings ψ<sub>''i''</sub>.]]
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.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ<sub>''i''</sub>, the open set condition requires that there exists a nonempty, open set
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.
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Taking [[natural logarithm]]s 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.<ref>{{Cite web | url=http://www.stat.uchicago.edu/~lalley/Papers/packing.pdf| title=The Packing and Covering Functions for Some Self-similar Fractals|last=Lalley|first=Steven|publisher=Purdue University|date=21 January 1988|access-date=2 February 2022}}</ref> 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.<ref>{{Cite web| url=http://users.jyu.fi/~antakae/publications/preprints/009-controlled_moran.pdf| title=Separation Conditions on Controlled Moran Constructions| last1=Käenmäki| first1=Antti| last2=Vilppolainen| first2=Markku| access-date = 2 February 2022}}</ref><ref>{{Cite journal| last=Schief| first=Andreas| title=Self-similar Sets in Complete Metric Spaces| journal=Proceedings of the American Mathematical Society| volume=124| issue=2| year=1996| url=https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03158-9/S0002-9939-96-03158-9.pdf}}</ref> In these cases, SOCS is indeed a stronger condition.
==See also==
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