A space is said to be power-homogeneous if some power of it is homogeneous. We prove that if a Hausdorff space
X
X
of point-countable type is power-homogeneous, then, for every infinite cardinal
τ
\tau
, the set of points at which
X
X
has a base of cardinality not greater than
τ
\tau
, is closed in
X
X
. Every power-homogeneous linearly ordered topological space also has this property. Further, if a linearly ordered space
X
X
of point-countable type is power-homogeneous, then
X
X
is first countable.