Collineation group as a subgroup of the symmetric group

Let $\mathrm{\Psi }$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group ${\mathfrak{S}}_{\mathrm{\Psi }}$ of the set $\mathrm{\Psi }$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $\mathrm{\Psi }$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $\mathrm{\Psi }$ is finite then $H$ contains the alternating subgroup ${\mathfrak{A}}_{\mathrm{\Psi }}$ of ${\mathfrak{S}}_{\mathrm{\Psi }}$. We show in Theorem \ref{density} below that $H={\mathfrak{S}}_{\mathrm{\Psi }}$, if $\mathrm{\Psi }$ is infinite.