Colored topological spaces? Let's say that for a topological space, we "color" it. By that, I mean we have some set of colors $C$, and we associate to each point in the space a color, and require continuous maps to preserve colors.
For example, we can color the faces of a polyhedron "white" and the edges "black". Two polyhedra are homeomorphic iff they are isomorphic as abstract polytopes.
My question is, has this concept (or a similar one) been defined before?

Yes indeed. The 8 faces of the regular octahedron may be colored alternately black and white, yielding the overall symmetry of the regular tetrahedron, which is a subgroup of the octahedral symmetry. By contrast a chequerboard appears at first sight to yield a symmetry subgroup of the square tiling, however it turns out to be the same symmetry but just referencing different elements of the tiling. For example twofold rotational symmetry is no longer about an edge mid-point but a vertex and mirror lines are spaced twice as far apart. Such twin-colorings of regular figures often turn out to be quasiregular (as in these two examples), but other colorings are possible.
Cromwell's Polyhedra includes an introduction to the topic in Chapter 9.
