About the properties of spaces $C_p(X)$ close to Frechet–Urysohn property

By analogy with the Fréchet-Urysohn property, the properties of $n$-Fréchet-Urysohn and $\omega$-Fréchet-Urysohn spaces of the spaces $C_p(X)$ are introduced into consideration. The connection between these properties and the properties $\gamma_n'$ and $\gamma_\omega'$ of the space $X$ is studied. In particular, it is established that the property $\gamma_\omega'$ of the space $X$ is equivalent the $\omega$-Frechet-Urysohn property of the space $C_p(X)$, and also that from the $n$-Frechet-Urysohn property it follows $\gamma_n'$.