Title: Chain Conditions in $C_p(X)$

Abstract: We present new results regarding calibers in function spaces $C_p(X)$. We calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $\lambda$-Lindel\"of extension of a discrete space of cardinality $\geq \lambda$. We give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X = \prod_{\xi < \lambda}X_\xi$ is a product of non-trivial Tychonoff spaces with $i$-weight less or equal to $\lambda$. The main theorem is: If $\kappa$ and $\lambda$ are cardinals with $\omega \leq \lambda\leq \kappa$, then the set of calibers of the space of the real-valued continuous functions defined on the one-point $\lambda$-Lindel\"of extension of the discrete space of cardinality $\kappa$ with its pointwise convergence topology, $C_p(L(\lambda, \kappa))$, is $\{\mu\in CN : cf(\mu)>\omega \ \wedge \ \mu\not\in (\lambda ,\kappa] \ \wedge \ cf(\mu)\not\in (\lambda,\kappa]\}$.