A symmetric d for a topological space R is said to be coherent if whenever
{
x
(
n
)
}
\{ x(n)\}
and
{
y
(
n
)
}
\{ y(n)\}
are sequences in R with
d
(
x
(
n
)
,
y
(
n
)
)
→
0
d(x(n),y(n)) \to 0
and
d
(
x
(
n
)
,
x
)
→
0
d(x(n),x) \to 0
, then
d
(
y
(
n
)
,
x
)
→
0
d(y(n),x) \to 0
. V. Niemytzki and W. A. Wilson have essentially shown that a topological space R is metrizable if and only if R is symmetrizable via a coherent symmetric. Conditions on a symmetric d which are equivalent to d being coherent are established. As a consequence, a theorem of A. Arhangel’skiĭ may be refined by showing that if
f
:
R
→
Y
f:R \to Y
is a quotient map from a metrizable space R onto a
T
0
{T_0}
-space y, then Y is metrizable if and only if f is a regular map.