$\bar S_i:H\r G$, $i=1\ld r$, satisfying the equality (5.3) for all $g\in H$. Then $r_{H,G}$ denotes the smallest $r$ as above.

Of course, every continuously perfect group is uniformly perfect.

Suppose that the closure of $B$ is included in $U$, where $B$ is a ball (or a half-ball and $n\geq 2$) and $U$ is open in $M$. Then $\H_B(M)$ is continuously perfect in $\H_U(M)$ with $r_{\H_B(M),\H_U(M)}=1$.

It suffices to observe that in the proof of Lemma 3.1 the homomorphism $S:\H_B(M)\r\H_U(M)$ is continuous, and the mapping $\bar S$ is a constant depending on $B$ and $U$.

The following fact is a consequence of Prop. 5.4.

If $\H_c(M)$ is continuously factorizable then it is also continuously perfect.

If $B_1\ld B_d\in\B$ is as in Def. 5.1(2), then choose any open subsets $U_1\ld U_d$ with $\overline B_i\s U_i$. Then we use Prop. 5.4 to each pair $(B_i,U_i)$.

However we do not know whether some homeomorphism groups $\H_c(M)$ are continuously factorizable. See also [@ryb] about locally continuously perfect groups of homeomorphisms.

Burago, Ivanov and Polterovich proved in the [@BIP] that $\D(M)$ is bounded (and a fortiori uniformly perfect) for many manifolds. We will need some preparatory notions and results from [@BIP]. A subgroup $H$ of 