An **acyclic space** can refer to several concepts depending on the context, but it is most commonly associated with graph theory and algebraic topology. 1. **In Graph Theory**: An acyclic graph (or directed acyclic graph, DAG) is a graph with no cycles, meaning there is no way to start at any vertex and follow a sequence of edges to return to that same vertex.
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A nonempty space is acyclic with the chosen coefficients if its reduced homology vanishes in every degree. Equivalently it has the homology of a point. It need not be contractible.