Equivalence relation/Symmetric accessibility relation/Example

Suppose that we are in a situation, where certain places (or Objects) can be reached from certain other places or not. This accessibility can be determined by the choice of means of transport, or by a more abstract kind of movement. Such an accessibility relation yields often an equivalence relation. A place can be reached by starting at this place, this gives reflexivity. The symmetry of accessibility means that if it is possible to reach ${}A$ starting from ${}B$, then it is also possible to reach ${}B$ starting from ${}A$. This is a natural condition for accessibility, though it is probably not fulfilled for every kind of accessibility. The transitivity does hold whenever we can do the movements after each other, like going from ${}A$ to ${}B$, and then from ${}B$ to ${}C$.

If, for example, accessible is given by walking overland, then two places are equivalent if and only if they lie on the same island (or continent). Islands and continents are the equivalence classes. In topology, the concept of path-connectedness plays an important role: two points are path-connected if they can be connected by a continuous path. Or: On the set of integer number, there is a colony of fleas, and every jump of a flea has the length of five unites (in both directions). How many flea populations are there, which fleas can meet each other?