We compare the magnetic properties of Co cluster assembled films with different degrees of oxidation. Clusters with grain size (2.3 +/- 0.7) nm are produced in a laser vaporization cluster source and soft-landed in ultrahigh vacuum conditions, forming highly porous nanogranular films. After exposure to air for different periods of time, the Co clusters oxidize and the sample may be considered as a thin antiferromagnetic Co oxide matrix containing ferromagnetic Co clusters. Magnetization measurements were performed in a temperature range from 300 down to 5 K, at applied magnetic fields up to 30 kOe. The exchange bias value at 5 K for the strongly oxidized sample is 4.8 kOe against the value of 0.75 kOe for the less oxidized sample. The mean values of the thicknesses of the Co oxide layers are estimated to be 0.6 and 0.3 nm for the more and less oxidized sample, respectively. We propose a method of measuring the exchange bias inducing temperature, i.e., the temperature at which exchange anisotropy is established. We determined the mean inducing temperatures for both samples, which are 55 and 25 K, respectively, for the more and less oxidized samples. Both temperatures are well below the bulk CoO Neel temperature of 292 K. A low value of the inducing temperature of the Co oxide layer is a consequence of its subnanometer thickness, while a large exchange bias value is a consequence of different dimensionality of Co clusters and Co oxide matrix.