  For an increasing sequence $(T_n)$ of one-parameter semigroups of sub
Markovian kernel operators over a Polish space, we study the limit semigroup
and prove sufficient conditions for it to be strongly Feller. In particular, we
show that the strong Feller property carries over from the approximating
semigroups to the limit semigroup if the resolvent of the latter maps the
constant 1 function to a continuous function.
  This is instrumental in the study of elliptic operators on $\mathbb{R}^d$
with unbounded coefficients: our abstract result enables us to assign a
semigroup to such an operator and to show that the semigroup is strongly Feller
under very mild regularity assumptions on the coefficients.
  We also provide counterexamples to demonstrate that the assumptions in our
main result are close to optimal.
