  Two sesquilinear forms $\Phi:\mathbb C^m\times\mathbb C^m\to \mathbb C$ and
$\Psi:\mathbb C^n\times\mathbb C^n\to \mathbb C$ are called topologically
equivalent if there exists a homeomorphism $\varphi :\mathbb C^m\to \mathbb
C^n$ (i.e., a continuous bijection whose inverse is also a continuous
bijection) such that $\Phi(x,y)=\Psi(\varphi (x),\varphi (y))$ for all $x,y\in
\mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizing
decomposition of a square complex matrix $A$; that is, a direct sum
$SAS^*=R\oplus J_{n_1}\oplus\dots\oplus J_{n_p}$, in which $S$ and $R$ are
nonsingular and each $J_{n_i}$ is the $n_i$-by-$n_i$ singular Jordan block. In
this paper, we prove that $\Phi$ and $\Psi$ are topologically equivalent if and
only if the regularizing decompositions of their matrices coincide up to
permutation of the singular summands $J_{n_i}$ and replacement of $R\in\mathbb
C^{r\times r}$ by a nonsingular matrix $R'\in\mathbb C^{r\times r}$ such that
$R$ and $R'$ are the matrices of topologically equivalent forms. Analogous
results for real and complex bilinear forms are also obtained.
