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.) For methods of turning a mathematical object into another one of the same kind, see the companion page on building complicated examples out of simple ones.