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Abstract A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $$\mathbb{R}^n$$ for some n , sending E to a starlike set. A subset $$E\subset X$$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $$\{E_i\}_{i=0}^{N+1}$$ such that $$E_{i}/E_{i+1}\subset X/E_{i+1}$$ is starlike-equivalent for each i and $$E_{N+1}$$ is a point. A decomposition $$\mathcal{D}$$ of a metric space X is said to be recursively starlike-equivalent, if there exists $$N\geq 0$$ such that each element of $$\mathcal{D}$$ is recursively starlike-equivalent of filtration length N . We prove that any null, recursively starlike-equivalent decomposition $$\mathcal{D}$$ of a compact metric space X shrinks, that is, the quotient map $$X\to X/\mathcal{D}$$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture.