A topological manifold is a topological space that locally resembles Euclidean space and is equipped with a topological structure. This means that around every point in the manifold, there exists a neighborhood that can be mapped to an open subset of Euclidean space, ensuring that the manifold behaves like familiar geometric spaces on a small scale. The concept connects deeply with how we understand charts, atlases, and smooth structures, forming a foundation for studying more complex shapes and spaces in mathematics.
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