# Space with finitely generated homology

From Topospaces

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 spacesView all homology-dependent properties of topological spaces OR view all homotopy-invariant properties of topological spaces OR view all properties of topological spaces

## Contents

## Definition

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.