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We give a $\delta$-constant criterion for equinormalizability of deformationsof isolated (not necessarily reduced) curve singularities over smooth basespaces of dimension $\geq 1$. For one-parametric families of isolated curvesingularities, we show that their topologically triviality is equivalent to theadmission of weak simultaneous resolutions.

Equinormalizability and topologically triviality of deformations of isolated curve singularities over smooth base spaces