<< 1. An n-dimensional manifold is a topological space whose points have neighborhoods homeomorphic to an n-dimensional ball. That's too technical, so I'll say the same thing without the jargon. 2. An n-dimensional manifold is such that every point has a region around it like an n-dimensional ball. Dropping the jargon makes it possible to equivocate about what "region around it" and "like ball" mean.

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The typical layperson still hasn't the foggiest notion of an n-dimensional ball for n > 3. --Dan