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But in a layperson's book by Penrose, he defined a manifold as a collection of points with a rule for telling you if a function defined on the manifold is smooth. He says this is more general than a collection of points with a local structure such as a metric tensor.

Can someone give me an example of a manifold defined in this more general way? I always thought smoothness required a distance function.

Also I'm also a little confused about the definition of smoothness. Penrose says a sphere is smooth, but a cube is not. I thought smoothness referred to functions defined on the manifold, not the manifold itself. Can't you have an unsmooth function defined on a smooth manifold (e.g., a function that goes to infinity on the sphere), or a smooth function defined on an unsmoothed manifold (e.g., the constant function on the cube).

Also, isn't the definition of a collection of points with a metric tensor also the same as the definition of a metric space? So what is the difference between a manifold and a metric space?