The Kuo condition, an inequality of Thom's type and \((c)\)-regularity (Q1373490)

In [\textit{K. Bekka}, Lect. Notes Math. 1462, 42-62 (1991; Zbl 0733.58003)] the first author introduced the notion of \((c)\)-regularity which is weaker than Whitney \((b)\)-regularity, and he showed that the \((c)\)-regularity condition implies topological triviality. In this paper, the authors give a characterization of \((c)\)-regularity (Theorem 2.4). Using it, they can show that the Kuo condition (given in Theorem 1.4) implies the \((c)\)-regularity condition (Theorems 2.7, 2.8). A different proof of the Trotman-Wilson result is also given (Corollary 2.9). Apart from this, the authors show that a particular Thom-type inequality is equivalent to another type of inequality (Theorem 2.14). Using this result, they show that the Thom condition in Theorem 1.3 is equivalent to the Kuiper-Kuo condition in the function case (Corollary 2.16). As another corollary of this, a result on Fukuda's ideal is obtained.