Natural topologies on Colombeau algebras

Abstract

We define intrinsic, natural and metrizable topologies ${\mathcal T}_{\Omega}$, ${\mathcal T}$, ${\mathcal T}_{s,\Omega}$ and ${\mathcal T}_s$ in ${\mathcal G}(\Omega)$, $\OK$, ${\mathcal G}_s(\Omega)$ and $\OK_s$, respectively. The topology ${\mathcal T}_{\Omega}$ induces ${\mathcal T}$, ${\mathcal T}_{s,\Omega}$ and ${\mathcal T}_s$. The topologies ${\mathcal T}_{s,\Omega}$ and ${\mathcal T}_s$ coincide with the Scarpalezos sharp topologies.