following

Let $M$ be a topological manifold and let $U$ be a neighborhood in $M$ of a compact subset $C$. For any neighborhood $\Q$ of the inclusion $i:U\subset M$ in $I(U;M)$ there are a neighborhood $\P$ of $i\in I(U;M)$ and a deformation $\varphi:\P\t I\r \Q$ into $I(U,C;M)$ which is modulo the complement of a compact neighborhood of $C$ in $U$ and such that $\varphi(i,t)=i$ for all $t$. We have also that if $D_i\subset V_i$, $i=1\ld q$, is a finite family of closed subsets $D_i$ with their neighborhoods $V_i$, then $\varphi$ can be chosen so that the restriction of $\varphi$ to $(\P\cap I(U,U\cap V_i;M))\t I$ assumes its values in $I(U,U\cap D_i;M)$ for each $i$.

Moreover, if $M$ has compact boundary $\pm$ then $\varphi$ restricted to $(\P\cap I(U,\pm\cap U; M))\t I$ takes its values into $I(U,\pm\cap U; M)$.

The first part coincides with Theorem 5.1[@ed-ki]. The second part is specified in Remark 7.2 in [@ed-ki].

We can derive from Theorem 2.2 the following fragmentation theorem.

Let $M$ be a compact manifold, possibly with boundary. Then the groups $\hm$, $\hdm$, $\P\hm$ and $\P\hdm$ are locally continuously factorizable, i.e. they satisfy Def. 2.1.

(See also [@ed-ki].) We will consider only the case of $\hm$, 