Let $f$, $g$ : $({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$ be two $C^\infty$ map-germs. Then $f$ and $g$ are $C^0$-equivalent if there exist homeomorphism-germs $h$ and $l$ of$({\Bbb R}^n,0)$ and $({\Bbb R}^p,0)$ respectively such that $g=l\circ f\circ h^{-1}$. Let$k$ be a positive integer.A germ $f$ is$k$-$C^0$-determined if every germ $g$ with $j^k g(0)=j^k f(0)$ is$C^0$-equivalent to $f$. Moreover, we say that $f$ isfinitely topologically determined if $f$ is$k$-$C^0$-determined for some finite $k$.We prove a theorem giving a sufficient condition for a germ to be finitely topologicallydetermined. We explain this condition below.

Let $N$ and $P$ be two $C^\infty$ manifolds. Consider the jet bundle $J^k(N,P)$with fiber $J^k(n,p)$. Let $z\in J^k(n,p)$ and let $f$ be such that $z=j^kf(0)$. Define\[\chi(f)=\dim_{\Bbb R} \frac{\theta (f)}{tf(\theta (n))+f^{\ast}(m_p)\theta(f)}.\]Whether $\chi(f)(k$ depends only on $z$, noton $f$ . We cantherefore define the set$W^k= W^k(n,p)=\{z\in J^k(n,p)\vert \chi(f)\ge k$for some representative $f$ of $z$\}.Let $W^k(N,P)$ bethe subbundle of $J^k(N,P)$ with fiber $W^k(n,p)$.Mather hasconstructed a finite Whitney (b)-regular stratification${\mathcal{S}}^k(n,p)$ of $J^k(n,p)-W^k(n,p)$ such that all strata are semialgebraic and$\mathcal K$-invariant, having the property that if${\mathcal S}^k(N,P)$ denotes the corresponding stratification of$J^k(N,P)-W^k(N,P)$ and $f\in C^\infty (N,P)$ is a $C^\infty$ map such that$j^k f$ is multitransverse to ${\mathcal S}^k(N,P)$, $j^k f(N)\capW^k(N,P)=\emptyset$ and $N$ is compact (or $f$ is proper), then $f$ istopologically stable.

For a map-germ $f:({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$, we define a certain {\L}ojasiewicz inequality. The inequality implies that there exists arepresentative $f : U\rightarrow{\Bbb R}^p$ such that $j^kf(U-\{0\})\cap W^k({\Bbb R}^n,{\Bbb R}^p)=\emptyset$ and such that $j^kf$ is multitransverse to${\mathcal S}^k({\Bbb R}^n,{\Bbb R}^p)$ at any finite set of points $S\subset U-\{0\}$.Moreover, the inequalitycontrols the rate $j^k f$ becomesnon-transverse as we approach 0. We show that if $f$ satisfies thisinequality, then $f$ is finitely topologically determined.

1991 Mathematics Subject Classification: 58C27.