On the Lindelof property of spaces of continuous functions over a Tychonoff space and its subspaces

We study relations between the Lindelof property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if C-p(X) is Lindelof, Y = X boolean OR {p}, and the point p has countable character in Y, then C-p(Y) is Lindelof; b) if Y is a cozero subspace of a Tychonoff space X, then l(C-p(Y)(omega)) <= l(C-p(X-omega) and ext(C-p(Y)(omega)) <= ext(C-p (X)(omega)).