I always thought one could define a manifold as a collection of points with a distance function or metric tensor. 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?