Context. My knowledge of geometry is pretty much zero. I wanted to know what a manifold is. (Just what the word means --- at least for now.) It seems a manifold is something that will locally feel like an Euclidean plane. So far so good. The Wikipedia article then gave examples of 1-manifolds such as a line and a circle, but then it said that a lemniscate is not a 1-manifold. The question is --- why not?
A wild guess. Looking at similar questions, I take the risk of saying that the problem is that a lemniscate crosses itself, so we could zoom in at a crossing point and the "x" from the crossing could never be "equivalent" ("homeomorphic" I guess) to an Euclidean line. Why not? Because the crossing point is never 1-dimensional --- it always needs two dimensions, otherwise it's not a crossing point.