Abstract. In his paper [10], Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is (a)-regular. Here we first discuss the Thom transversality theorem for the weak topology and then give a similiar kind of result for the weak topology, under very weak hypotheses. Recently several transversality theorems have been proved for complex manifolds and holomorphic maps (see [7] and [3]). In view of these transversality theorems we also prove a result analogous to Trotman's result in the complex case.
Smooth caseLet M and N be smooth manifolds. Denote by C ∞ (M, N ), the set of all smooth maps between M and N . The set C ∞ (M, N ) can then be given two topologies, the weak topology and the strong (Whitney) topology (see page 35 in Hirsch [6] for the definitions, see also [2] for more about the function space topologies). Many authors prefer to use a definition of this topology on the function spaces via jets; see page 42 in Golubitsky and Guillemin [4] and also a somewhat detailed discussion about the different topologies on function spaces in du Plessis and Vosegaard [1]. We will follow the approach of Hirsch [6]. Denote by C ∞ W (M, N ) and C ∞ S (M, N ), the space of smooth maps between M and N with the weak topology and the strong topology respectively. We prove the results for the weak topology and state them for the strong topology.