Is there standard mathematical terminology for structures that come in one-to-one pairs?

Question

Background / concrete example of what I'm asking about:

Given a set $X$ and a set of "open" sets $\mathcal T\subseteq\mathcal P(X)$ satisfying the open-set axioms (of topological spaces), we can immediately define a closure operator $\text{cl}:\mathcal P(X)\ni A\mapsto\text{cl}(A):=\bigcap\{C\supseteq A\,|\,X\setminus C\in\mathcal T\} \in\mathcal P(X)$, and then we can prove that cl satisfies Kuratowski's closure axioms.

On the other hand, given a set $X$ and operator $\text{cl}:\mathcal P(X)\to\mathcal P(X)$ satisfying the closure axioms, we can define a subset $A\subseteq X$ to be "closed" iff $\text{cl}(A)=A$, and "open" iff $X\setminus A$ is closed. Then we can prove the set $\mathcal T$ of open subsets of $X$ satisfies the open-set axioms for a topological space.

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Question:

I've heard it said (and I've probably said) that the above constructions are two "equivalent" ways of defining topological spaces. But this "equivalence" is not in the usual sense, where propositions $A$ and $B$ are "equivalent" if $A$ implies $B$ and also $B$ implies $A$. Rather, we have two completely different objects (a set of open subsets, and a closure operator); given one object, there is a canonical way to construct the other. And if given object $A$ (of type "a") I construct object $B$ (of type "b"), then using the reverse-direction construction on object $B$ gives back object $A$.

There is a clear sense in which these constructions are "equivalent," but I'm confused about how to talk about this. Moreover, this seems like the kind of general, abstract situation that mathematicians have precise terminology for. Maybe open-set topologies and closure operators come in "dual pairs"? Or "the class of "(open-set topologies) $\cup$ (closure operators)" is partitioned into associated open-set-topology/closure-operator pairs"? These sound clunky and unclear, so I'm curious, is there a better (and generalizable) way to describe this phenomenon, where two different types of objects come in associated pairs?

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Here's one more example, to show how this idea generalizes. In a Banach space $X$, given any projector $P$, there is an associated decomposition $X=\text{Range}(P)\oplus\text{Range}(I-P)$, where the two subspaces are closed. Likewise, for each decomposition of the form $X=U\oplus V$ where $U$ and $V$ are closed subspaces, there are associated projectors $P$ and $I-P$ such that $\text{Range}(P)=U$ and $\text{Range}(I-P)=V.$ So pairs of projectors of the form $P$ and $I-P$ are associated uniquely to pairs of closed subspaces of $X$, and this "pairing" is one-to-one.

But pairs of projectors and pairs of closed subspaces are not the same type of object. It would be nice if we could succinctly describe this situation with language similar to that of the previous example.

Answer 1

These seem to be examples of (bi-interpretable) synonymy, or possibly the weaker bi-interpretability. In When Bi-Interpretability Implies Synonymy, Friedman and Visser define both synonymy (originally from de Bouvre) and bi-interpretability (originally from Alhbrandt and Ziegler):

Two theories $U$ and $V$ are synonymous iff there is a theory $W$ that is both a definitional extension of $U$ and of $V$.

Equivalently, $U$ and $V$ are synonymous iff there are interpretations [nlab] $K : U \rightarrow V$ and $M : V → U$, such that $V$ proves that the composition $K \circ M$ is the identity interpretation on $V$ and such that $U$ proves that the composition $M \circ K$ is the identity interpretation on $U$.

Two theories $U$ and $V$ are bi-interpretable iff there are interpretations $K : U → V$ and $M : V → U$, such that there is a $V$-definable function $F$, such that $V$ proves that $F$ is an isomorphism between $K \circ M$ and the identity interpretation on $V$ and such that there is a $U$-definable function $G$, such that $U$ proves that $G$ is an isomorphism between $M \circ K$ and the identity interpretation on $U$.

I believe the following quote from the OP shows that the examples satisfy the second definition of synonymy.

Rather, we have two completely different objects (a set of open subsets, and a closure operator); given one object, there is a canonical way to construct the other. And if given object A (of type "a") I construct object B (of type "b"), then using the reverse-direction construction on object B gives back object A.

$U$ and $V$ are the theories with either the open-set axioms or Kuratowski's closure axioms, and the interpretations are given from the canonical construction. What might be troublesome is showing that the composition is the identity - I haven't checked, but there might be subtle encoding issues, which would lead to the theories only being bi-interpretable.