Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain cases, this lead to new manifolds which could not be realized as subsets of euclidean spaces. Eg non-orientable surfaces cannot be embedded in $R^3$, non-algebraic manifolds etc.

Nowadays, fractals are found as iterations of certain polynomial maps on euclidean space. (or some other ways, as wikipedia says, but still embdedded). Are we missing out on some fractals? Is there an intrinsic definition of fractals? (This would also have to come with an intrinsic definition of dimension. Haudsorff dimension is still an embedded dimensions, using covering of balls in euclidean space.)

specificeuclidean space (which is the case for the Klein bottle and $\mathbb{R}^3$, mentioned in the first paragraph of the question)... $\endgroup$ – Qfwfq Jun 1 '18 at 17:18