In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them?
Edit:
To clarify, I am considering these spaces - Chebyshev, discrete, maybe a Paris space (wiki).
I define conic sections via their distance definition, i.e. for example ellipse is defined as $d(X, F_1) + d(X, F_2) = 2a$, and then I try to get the shape of it from definitions of metric.