Space with finitely generated homology
This article defines a property of topological spaces that depends only on the homology of the topological space, viz it is completely determined by the homology groups. In particular, it is a homotopy-invariant property of topological spaces
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A topological space is said to have finitely generated homology if it has only finitely many nonzero homology groups, and each of them is a finitely generated group. In other words, the Betti numbers are all finite and only finitely many of them are nonzero.