Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. [CH Dowker](http://en.wikipedia.org/wiki/Clifford_Hugh_Dowker) constructed two simplicial complexes $K$ and $L$ associated to $R$:


1. a simplex in $K$ consists of finitely many elements $x \in X$ such that there exists a single $y \in Y$ with $(x,y) \in R$, and
2. a simplex in $L$ consists of finitely many elements $y \in Y$ such that there exists a single $x \in X$ with $(x,y) \in R$.


Clearly, these are simplicial complexes. The main theorem of Dowker is the construction of a natural isomorphism of homology groups of $K$ and $L$. There is even a proof outline on [nlab](http://ncatlab.org/nlab/show/Dowker%27s+theorem). In fact there is a homotopy equivalence between the geometric realizations although that requires an ordering on the simplices and is therefore not natural. This may be found in the paper



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> **C. H. Dowker**, *[Homology Groups of Relations](http://www.maths.ed.ac.uk/~aar/papers/dowker.pdf)*, Annals of Math, 56, (1952), 84–95.
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The goal of this paper was to prove equivalence of Cech, Vietoris and Alexander (co)homology theories. My question is 



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> What other applications (if any) have been found for this theorem?
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In particular, the fact that the relation itself creates the simplicial complexes seems to make this theorem have very limited uses beyond the consequences proved already by Dowker: I can't seem to find much on the internet at least. I wonder if the experts have some idea...



What are the applications of Dowker's theorem?
There is a considerable literature on `applications' of Dowker's result to sociology. This is sometimes doubtful in its depth! The development was started by R. Atkin. 


As an example look at 


<http://www.ehu.es/ccwintco/uploads/1/11/Blanca-Cases-Q-analysis.pdf>


As an offshoot of this there is fairly recent work in discrete maths (see work by Hélène Barcelo). I will not try to describe this other than saying it looks at an idea of the connectivity of a relation.


Back in the world of algebraic topology, it provides a way of proving that the pro-object in the homotopy category of simplicial sets, that is given by the Cech complex construction is in fact homotopy coherent. This provides a way of linking strong shape theory to the original form of shape theory. (It is not hard to prove this coherence directly although I only know one proof that has been written down in a thesis of one of my ex-students.)


(Edit: I forgot another example. You start with a group, $G$, and a family of subgroups, and ask to what extent invariants of the family give you invariants of the big group. This was the subject of a paper by Abels and Holz (Higher generation by subgroups , J. Alg, 160, (1993), 311– 341.) The family generates a covering of $G$ by its cosets. The two complexes given by that covering allow proof to be shortened and also in certain circumstances for links to Volodin's alegrbaic K-theory to be given.)


