I suppose so, but they really are trying to convey that all the points lie close to some lower-dimensional crumpled surface. So a subset that was e.g. just a la

I suppose so, but they really are trying to convey that all the points lie close to some lower-dimensional crumpled surface. So a subset that was e.g. just a lattice in the high-dimensional space wouldn't fit this mental image. E.g. Generate 3d points that are on a 2d plane +- some small random offset normal to the plane. If the points are isolated, without each having a local neighbourhood, it's not technically a manifold. But, you could describe the data-set as lying on or near a plane to within some tolerance. So they're trying to describe something that is much more specific / strongly constrained than an arbitrary subset, but it doesn't meet the very stringent (and frankly idealized) requirements of a manifold. I wonder whether there is a math-object to describe this? EDIT> Maybe it's just a matter of saying "close to some lower-dimensional manifold", rather than "on a lower-dimensional manifold."