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.)



Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?
Topological dimension (say, covering dimension) $\dim\_\mathrm{T}$ and Hausdorff dimension $\dim\_\mathrm{H}$ both make sense for metric spaces. **Benoit Mandelbrot** defined $A$ to be a *fractal* iff $\dim\_\mathrm{T} A < \dim\_\mathrm{H} A$. The packing dimension $\dim\_\mathrm{P}$ also makes sense in metric space. **James Taylor** defined $A$ to be a *fractal* iff $\dim\_\mathrm{H} A = \dim\_\mathrm{P} A$. Also making sense for metric space is the definition of **Michael Barnsley** ... A *fractal* is an element of the hyperspace $\mathbb{H}(K)$ of a compact metric space $K$. Perhaps you have your own, different, definition? 


Definitions for all these are in my book *Integral, Probability, and Fractal Measures*


