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

summary:We study relations between the Lindelöf 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 Lindelöf, $Y=X\cup\{p\}$, and the point $p$ has countable character in $Y$, then $C_p(Y)$ is Lindelöf; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l(C_p(Y)^\omega)\le l(C_p(X)^\omega)$ and $\operatorname{ext}(C_p(Y)^\omega)\le \operatorname{ext}(C_p(X)^\omega)$