in some cases, only a dense ) subset of the set of mappings. To make such a statement precise, it is necessary to define the space of mappings under consideration, and what is the topology in it. There are several possibilities; see the book by Hirsch. What is usually understood by Thom's transversality theorem is a more powerful statement about jet transversality. See the books by Hirsch and by Golubitsky and Guillemin. The original reference is Thom, Bol. Soc. Mat. Mexicana (2) 1 (1956), pp. 59–71. John Mather proved in the 1970s an even more general result called the multijet transversality theorem. See the book by Golubitsky and Guillemin. Infinite-dimensional version The infinite-dimensional version of the transversality theorem takes into account that the manifolds may be modeled in Banach spaces. Formal statement Suppose is a map of -Banach manifolds. Assume: (i) and are non-empty, metrizable -Banach manifolds with chart spaces over a field (ii) The -map with has as a regular value. (iii) For each parameter , the map is a Fredholm map, where for every (iv) The convergence on as and for all implies the existence of a convergent subsequence as with If (i)-(iv) hold, then there exists an open, dense subset such that is a regular value of for each parameter Now, fix an element If there exists a number with for all solutions of , then the solution set consists of an -dimensional -Banach manifold or the solution set is empty. Note that if for all the solutions of then there exists an open dense subset of such that there are at most finitely many solutions for each fixed parameter In addition, all these solutions are regular. References Theorems in differential topology Differential geometry