A topological -manifold is a Hausdorff second-countable space in which every point has a neighborhood homeomorphic to an open subset of .
An -manifold has zero singular homology in every degree greater than . One proof uses a countable exhaustion by subsets that retract onto -dimensional CW complexes and the compact support of every singular cycle.
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A **topological manifold** is a fundamental concept in topology and differential geometry. It is a topological space that, in informal terms, resembles Euclidean space locally around each point.