This is the definition given by most authors (Kelley 1955, p. 117; Willard 1970, pp. 94-95). However, some authors (e.g., Cullen 1968, p. 130) require the
additional condition that be a T1-space. In any case, every
completely regular space is regular, and the converse is not true.

Cullen, H. F. "Normal Spaces. Completely Regular Spaces." §18 in Introduction
to General Topology. Boston, MA: Heath, pp. 118-140, 1968.Joshi,
K. D. Introduction
to General Topology. New Delhi, India: Wiley, p. 163, 1983.Kelley,
J. L. General
Topology. New York: Van Nostrand, 1955.Willard, S. "Regularity
and complete regularity." §14 in General
Topology. Reading, MA: Addison-Wesley, pp. 92-99, 1970.