Any compact first countable space X possesses a base B such that the family \(P_ B=\{Fr(U):\) \(U\in B\}\) has the order less than \({\mathfrak C}\) at every \(x\in X\). Therefore CH implies X has a peripherally point- countable base. We also prove that every first countable compact space with a peripherally point-countable base is a continuous image of a zero- dimensional first countable compact space, giving thus a new easier way to prove A. V. Ivanov’s theorem [Usp. Mat. Nauk 35, No.6(216), 161- 162 (1980; Zbl 0458.54021)].