I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are

1. Topological Spaces. These can be defined in terms of open sets, closed sets, neighbourhoods, the Kuratowski closure axioms, etc.

2. Matroids. These can be defined via independent sets, bases, circuits, rank functions, etc.

Are there are other good examples?

Secondly, what are some advantages of multiple axiomatizations?

Obviously, one advantage is that one can work with the most convenient definition depending on the task at hand. Another is that they allow different generalizations of the object in question. For example, infinite matroids can be axiomatized by adapting the independent set axioms, but it is unknown how to axiomatize them via the circuit axioms. An acceptable answer to the second question would be an example of a proof in one axiom system that doesn't translate easily (not sure how to make this precise) into another axiom system.