How should we prove equivalence of definitions?

Question

As the title says, if we happen to see two different definitions (perhaps from different sources) for the same object or concept, how should we prove that the two definitions are equivalent?

To be more precise, I give an example I have in mind:

https://en.wikipedia.org/wiki/Characterizations_of_the_category_of_topological_spaces

Now, this is not like whether we define boundary of a set as "closure minus interior" or "intersection of closure and closure of complement", because both are sets and I think there is not much argument on how we should show two sets to be equal.

In the example in the link, the structures are actually quite different. Open sets are not interior operators, open sets are not all of the neighbourhoods, and neighbourhoods may not always be open... The only thing we can do is to recover other structures from one framework, for instance, we define what a neighbourhood is provided that open sets are taken as axioms.

To prove that those definitions are defining the same concept, I think we first need to show that given any of those frameworks, we can recover other structures in the above sense. This part should be easy. Now, given a topology (defined by open sets), we can recover the neighbourhood function, and with that neighbourhood function, we can recover open sets again.

However, there is no guarantee that the open sets obtained are actually the original open sets, and I suppose we need to prove for this, that the open sets are the original ones. Similarly, if we start from a neighbourhood function, recover the open sets, and recover the neighbourhood function again, I think we actually need to prove that the neighbourhood function obtained is the original one.

To make things even more complicated, we should even consider whether "diagrams commute" in the sense that if we start from definition A (say open sets), generate structure B (say neighbourhood function) directly, or we take another path by generating structures C, and then D, ... (say, closure operator, interior operator, ...) and then B at last, do I actually obtain the same structure B (same neighbourhood function)?

I think we do need to prove the above, that the process of obtaining other structures are reversible and the "diagram commutes". Correct me if I am wrong. But then we are in big trouble. Although there are finitely many definitions and the number of paths is just finite, I don't think a mathematician would just sit there for a week just to prove that they are indeed the same. Is that we don't need to prove the above, or we do need to prove the above, but there is some way more efficient than proving for all possible paths one by one?

Answer 1

There is no need to go through "paths". The best example I have in mind is that of Boolean algebra. Sometimes it's defined as a ring with special properties, and sometimes as a special kind of lattice. But they're equivalent definitions, because of the following. Say we differentiate them as Boolean ring on the one hand, Boolean lattice on the other. Then there is a construction, say $F$ that turns the boolean ring $B$ into a Boolean lattice $FB$. Conversely, there is a construction $G$, that turns the Boolean lattice $L$ into a Boolean ring $GL$. Now what gives the equivalence, is that for any Boolean ring $B$, $GFB$ is isomorphic (or even equal in this case) to $B$, and for any Boolean lattice $L$, $FGL$ is isomorphic to $L$. This simply allows us to work with both frameworks at once, and jump from one to the other, knowing that there is no risk. The same goes for topological spaces : to any Open-space (a set together with open sets) you can associate a Neighbourhoodfunction-space, and vice versa, such that if we let $F$, $G$ denote the two constructions, then $FG = Id$ and $GF= Id$. For any "path" of equivalent constructions you can take (if you take equivalent to mean what I've described), then it will change nothing. If you want a rigourous, formalized version of this, then you can take a closer look at category theory and see that I've just described inverse pairs of functors