It is often possible to prove that an object of type X exists with certain properties exists by building it out of a suitable object of type Y. This page contains links to several other pages that discuss this phenomenon for various different choices of X and Y. (There are many other circumstances under which one might wish to build one kind of mathematical object out of another, but the articles below are focused on existence theorems.)

Turning groups into algebras

Turning topological spaces into algebras

Using ordinals to build Banach spaces

Algebraic constructions of graphs

Geometrical constructions of graphs

Using polynomials to build manifolds

Algebraic constructions of sets of integers with given properties

Building topological spaces out of groups