Suppose there is a set of objects on which we can define an equivalence relation. Under some transformations of the space on which the objects are defined, these objects may change their equivalence class. As a trivial example consider vectors on a $2$ dimensional euclidean space and the equivalence relation defined by $a \sim b$ if their $x$-components have the same sign or they are both null. In this example there are $3$ equivalence classes, but under a change of coordinates one vector may transition from one equivalence class to another. In this case I suppose the equivalence relation should be defined on each particular coordinate system.
I want to find out whether there may be any interesting properties in such cases. Could anyone provide some insight or literature on this subject if there is any? I could not find anything relevant on this topic.