In a previous paper, D. W. Hajek showed that if a space
X
X
is a
T
3
{T_3}
space and
A
A
is a compact subset of
W
X
WX
, the Wallman compactification of
X
X
, then
X
∩
A
X \cap A
is a closed subset of
X
X
. This raises the question of whether this “closed intersection” property characterizes the
T
3
{T_3}
spaces among the Hausdorff spaces. In the present paper, the authors show this conjecture is false by giving an example of a nonregular Hausdorff space whose Wallman compactification is a
KC
\operatorname {KC}
(compact closed)-space, and, hence, trivially satisfies this “closed intersection” property.