The eROSITA Final Equatorial-Depth Survey (eFEDS): Identification and characterization of the counterparts to the point-like sources

In Fall 2019, eROSITA on board of SRG observatory started to map the entire sky in X-rays. After the 4-year survey program, it will reach flux limits about 25 times deeper than ROSAT. During the SRG Performance Verification phase, eROSITA observed a contiguous 140 deg^2 area of the sky down to the final depth of the eROSITA all-sky survey ("eROSITA Final Equatorial-Depth Survey": eFEDS), with the goal of getting a census of the X-ray emitting populations that will be discovered over the entire sky. This paper presents the identification of the counterparts to the point-sources detected in eFEDS in the Main and Hard samples described in Brunner et al., and their multi-wavelength properties, including redshift. For the identification of the counterparts we combined the results from two independent methods NWAY and ASTROMATCH, trained on the multi-wavelength properties of a sample of 23k XMM-Newton sources detected in the DESI Legacy Imaging Survey DR8. Then spectroscopic redshifts and photometry from ancillary surveys are collated for the computation of photometric redshifts. The eFEDS sources with a reliable counterparts are 24774/27369 (90.5\%) in the Main sample and 231/246 (93.9\%) in the Hard sample, including 2514 (3) sources for which a second counterpart is equally likely. By means of reliable spectra, Gaia parallaxes, and/or multiwavelength properties we have classified the counterparts in both samples as 'Galactic' (2822) and 'extragalactic' (21952). For about 340 of the extragalactic sources we cannot rule out the possibility that they are unresolved clusters or belong to clusters. Inspection of the distributions of the X-ray sources in various optical/IR color-magnitude spaces reveal a rich variety of diverse classes of objects. The photometric redshifts are most reliable within the KiDS/VIKING area, where also deep near-infrared data is available.(abridged)