An EUV collector for collecting and transmitting radiation from an EUV radiation source includes at least one collector mirror for reflecting an emission of the EUV radiation source, which is rotationally symmetric with respect to a central axis. The EUV collector also includes a cooling device for cooling the at least one collector mirror. The cooling device has at least one cooling element, which has a course with respect to the collector mirror, in each case, such that the projection of the course into a plane perpendicular to the central axis has a main direction, which encloses an angle of at most 20° with respect to a predetermined preferred direction. The collector transmits improved quality radiation to illuminate an object field.