---
type: Fractal Concept
title: Open Set Condition
description: 'Separation hypothesis on an iterated function system: there exists a nonempty bounded open set U with the union of the maps'' images of U contained in U and those images pairwise disjoint.'
resource: https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/open-set-condition/
tags:
- dimension-theory
- measure-theory
aliases:
- osc
- moran condition
- moran open set condition
sources:
- id: glossary
  resource: https://tyson-swetnam.github.io/fractal-notebooks/foundations/glossary/
  title: Fractal Notebooks glossary (Chapter 4)
- id: seed
  resource: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/fractal_rag/src/fractal_rag/concepts/seed.yaml
  title: fractal_rag concept seed (seed.yaml)
generated:
  by: process:fractal-rag-okf-export
  at: "2026-10-24T00:00:00Z"
status: stable
---

# Definition

Separation hypothesis on an iterated function system: there exists a nonempty bounded open set U with the union of the maps' images of U contained in U and those images pairwise disjoint. When it holds for similarities, the similarity dimension log N / log(1/r) equals the Hausdorff dimension (Moran 1946; Hutchinson 1981). It is the precise condition that licenses the closed-form dimension formulas for the classical self-similar fractals.

# Aliases

`osc`, `moran condition`, `moran open set condition`

# Related concepts

* [Iterated Function System](../ifs/iterated-function-system.md)
* [Hausdorff Dimension](hausdorff-dimension.md)
* [Self-Similarity](../geometry/self-similarity.md)
* [Fractal](../geometry/fractal.md)

# Citations

[1] [Fractal Notebooks glossary (Chapter 4)](https://tyson-swetnam.github.io/fractal-notebooks/foundations/glossary/) — `docs/foundations/glossary.md`

[2] [fractal_rag concept seed (seed.yaml)](https://github.com/tyson-swetnam/fractal-notebooks/blob/main/fractal_rag/src/fractal_rag/concepts/seed.yaml) — `fractal_rag/src/fractal_rag/concepts/seed.yaml`