Collineation group as a subgroup of the symmetric group
Let Ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group GΨ of the set Ψ. Suppose that H contains the projective group and an arbitrary self-bijection of Ψ transforming a triple of collinear points to a non-collinear triple. It is well-known from  that if Ψ is finite then H contains the alternating subgroup AΨ of GΨ.
We show in Theorem 3.1 below that H = GΨ, if Ψ is infinite.