Example

Let $C$ be a $k$-algebra and $N$ a nilpotent two-sided ideal of $C$. The proposition applies to the algebra $C$ endowed with the $N$-adic filtration. If $A$ is a formally smooth linearly topologized $k$-algebra, one obtains that every continuous homomorphism of $A$ into the $k$-algebra $C/N$, endowed with the discrete topology, lifts to a continuous homomorphism of $A$ into the $k$-algebra $C$, endowed with the discrete topology.