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. 



Cryptomorphisms
The phenomenon that I think you have in mind has a name: **cryptomorphism**. I learned the name from the writings of Gian-Carlo Rota; Rota's favorite example was indeed matroids. Gerald Edgar informs me that the name is due to Garrett Birkhoff.


I think modern mathematics is replete with cryptomorphisms. In my class today, I presented the "Omnibus Hensel's Lemma". Part a) was: the following five conditions on a valued field are all equivalent. Part b) was: complete fields satisfy these equivalent properties. There are lots more equivalent conditions than the five I listed: see 


[An unfamiliar (to me) form of Hensel's Lemma](https://mathoverflow.net/questions/15673/an-unfamiliar-to-me-form-of-hensels-lemma)


and especially Franz Lemmermeyer's answer for further characterizations. 


I would say that the existence of cryptomorphisms is a sign of the richness and naturality of a mathematical concept -- it means that it has an existence which is independent of any particular way of thinking about it -- but that on the other hand the existence of not obviously equivalent cryptomorphisms tends to make things more complicated, not easier: you have to learn several different languages at once. For instance, the origin of the question I cited above was the fact that in Tuesday's class I st*p*dly chose the wrong form of Hensel's Lemma to use to try to deduce yet another version of Hensel's Lemma: it didn't work! Since we are finite, temporal beings, we often settle for learning only some of the languages, and this can make it harder for us to understand each other and also steer us away from problems that are more naturally phrased and attacked via the languages in which we are not fluent. Some further examples:


I think that the first (i.e., most elementary) serious instance of cryptomorphism is the determinant. Even the Laplace expansion definition of the determinant gives you something like $n$ double factorial different ways to compute it; the fact that these different computations are not obviously equivalent is certainly a source of consternation for linear algebra students. To say nothing of the various different ways we want students to think about determinants. It is "just" the signed change of volume of a linear transformation in Euclidean space (and the determinant over a general commutative ring can be reduced to this case). And it is "just" the induced scaling factor on the top exterior power. And it is "just" the unique scalar $\alpha(A)$ which makes the adjugate equation $A\*\operatorname{adj}(A) = \alpha(A) I\_n$ hold. And so forth. You have to be fairly mathematically sophisticated to understand all these things.


Other examples: 


Nets versus filters for convergence of topological spaces. Most standard texts choose one and briefly allude to the other. As G. Laison has pointed out, this is a disservice to students: if you want to do functional analysis (or read works by American mathematicians), you had better know about nets. If you want to do topological algebra and/or logic (or read works by European mathematicians), you had better know about filters.


There are (at least) three axiomatizations of the concept of **uniform space**: (i) entourages, (ii) uniform covers, (iii) families of pseudometrics. One could develop the full theory using just one, but at various points all three have their advantages. Is there anyone who doesn't wish that there were just one definition that would work equally well in all cases?


