We present the analysis of ROSAT HRI and PSPC X-ray data on the compact cD type cluster Abell 4059 .
The central cD galaxy hosts the strong radio source PKS2354-35 .
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 PKS2354-35 lie perpendicularly to the X-ray bar and extend into the two X-ray minima .
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 which extends far enough to fill the two X-ray holes .
An alternative scenario for the anti-correlated radio and X-ray structure is an rotating disk in the cooling flow region of Abell 4059 .

The ROSAT HRI surface brightness profile shows that there is a cooling flow around the central cD galaxy with a cooling rate of \dot { M } = 184 ^ { +22 } _ { -25 } M _ { \odot } 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 rate of \dot { M } < 80 M _ { \odot } yr ^ { -1 } ( 90 % confidence level ) .
The X-ray spectra of the central region indicate very little intrinsic absorption in the cluster , with an upper limit of \Delta N _ { H } \leq 4.0 \times 10 ^ { 19 } cm ^ { -2 } for excess absorption in front of the cluster emission at the center of the cooling flow .

On large scales , the cluster shows interesting spatial alignment between the major axes of the X-ray emission , cD galaxy , and the cluster potential .
This is consistent with the anisotropic merger model for the formation of cD galaxies and the orientation of cD galaxy radio sources .