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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