We present the analysis of ROSAT HRI and PSPC X-ray data on the compac
t cD type cluster Abell 4059. The central cD galaxy hosts the strong r
adio source PKS 2354-35. The central portion of the X-ray image shows
two statistically significant X-ray emission minima and an orthogonal
X-ray bar. As is the case in NGC 1275, the radio lobes in PKS 2354-35
lie perpendicularly to the X-ray bar and extend into the two X-ray min
ima. One suggestion is that the radio lobe plasma has displaced the X-
ray plasma and created the X-ray holes. The radio plasma and the X-ray
gas do have comparable pressures in the radio lobe area. However, the
available radio images do not show radio emission that extends far en
ough to hh the two X-ray holes. An alternative scenario for the antico
rrelated radio and X-ray structure is an rotating disk in the cooling
how region of Abell 4059. The ROSAT HRI surface brightness profile sho
ws that there is a cooling flow around the central cD galaxy with a co
oling rate of M = 184(-25)(+22) M-circle dot yr(-1) within a radius of
156 kpc. However, the ROSAT PSPC spectrum of the central regions does
not require a cooling flow and gives an upper limit to the cooling ra
te of M < 80 M-circle dot yr(-1) (90% confidence level). The X-ray spe
ctra of the central region indicate very little intrinsic absorption i
n the cluster, with an upper limit of Delta N-H less than or equal to
4.0 x 10(19) cm(-2) for excess absorption in front of the cluster emis
sion at the center of the cooling flow. On large scales, the cluster s
hows interesting spatial alignment between the major axes of the X-ray
emission, cD galaxy, and the cluster potential. This is consistent wi
th the anisotropic merger model for the formation of cD galaxies and t
he orientation of cD galaxy radio sources.