In this paper, we show that: (1) every strongly ${omega}$-monolithic space X with countable fan-tightness is Fr$'{e}$chet-Urysohn; (2) a direct proof of that X is Lindel$"{o}$f when $C_p$(X) is Fr$'{e}$chet-Urysohn; and (3) X is Lindel$"{o}$f when X is paraLindel$"{o}$f and $C_p$(X) is AP. (3) is a generalization of the result of [8]. And we give two questions related to Fr$'{e}$chet-Urysohn and AP properties on $C_p$(X).