A multi-stage compressor (2,102), comprising: a plurality of compression chambers (40,41) formed in a housing; a medium-pressure refrigerant discharge port (23,123) that discharges medium-pressure refrigerant from a low-stage compression chamber of the compression chambers (40,41); a medium-pressure refrigerant suction port (22,122) that is open in a same direction as the medium-pressure refrigerant discharge port (23,123) and induces the medium-pressure refrigerant into a high-stage compression chamber of the compression chambers (40,41); a high-pressure refrigerant discharge port (24,124) that is open in a same direction as the medium-pressure refrigerant discharge port (23,123) and discharges high-pressure refrigerant discharged from the high-stage compression chamber of the compression chambers (40,41); and a refrigerant connection cover that is detachably mounted on the housing, wherein the refrigerant connection cover forms a medium-pressure refrigerant chamber, which communicates with the medium-pressure refrigerant suction port (22,122) and the medium-pressure refrigerant discharge port (23,123) and which has an external medium-pressure refrigerant connection induction port that is open toward an outside, wherein the refrigerant connection cover forms a high-pressure refrigerant chamber, which communicates with the high-pressure refrigerant discharge port (24,124) and which has a high-pressure refrigerant ejection port open toward an outside, wherein the multi-stage compressor (2,102) is a scroll compressor including an orbiting scroll and a fixed scroll, and the fixed scroll constitutes a part of the housing, and the refrigerant connection cover is mounted on the fixed scroll, wherein the medium-pressure refrigerant chamber includes a recess portion of the fixed scroll and a recess portion of the refrigerant connection cover that are facing each other, wherein the high-pressure refrigerant chamber includes a recess portion of the fixed scroll and a recess portion of the refrigerant connection cover that are facing each other. a plurality of compression chambers (40,41) formed in a housing; a medium-pressure refrigerant discharge port (23,123) that discharges medium-pressure refrigerant from a low-stage compression chamber of the compression chambers (40,41); a medium-pressure refrigerant suction port (22,122) that is open in a same direction as the medium-pressure refrigerant discharge port (23,123) and induces the medium-pressure refrigerant into a high-stage compression chamber of the compression chambers (40,41); a high-pressure refrigerant discharge port (24,124) that is open in a same direction as the medium-pressure refrigerant discharge port (23,123) and discharges high-pressure refrigerant discharged from the high-stage compression chamber of the compression chambers (40,41); and a refrigerant connection cover that is detachably mounted on the housing, wherein the refrigerant connection cover forms a medium-pressure refrigerant chamber, which communicates with the medium-pressure refrigerant suction port (22,122) and the medium-pressure refrigerant discharge port (23,123) and which has an external medium-pressure refrigerant connection induction port that is open toward an outside, wherein the refrigerant connection cover forms a high-pressure refrigerant chamber, which communicates with the high-pressure refrigerant discharge port (24,124) and which has a high-pressure refrigerant ejection port open toward an outside, wherein the multi-stage compressor (2,102) is a scroll compressor including an orbiting scroll and a fixed scroll, and the fixed scroll constitutes a part of the housing, and the refrigerant connection cover is mounted on the fixed scroll, wherein the medium-pressure refrigerant chamber includes a recess portion of the fixed scroll and a recess portion of the refrigerant connection cover that are facing each other, wherein the high-pressure refrigerant chamber includes a recess portion of the fixed scroll and a recess portion of the refrigerant connection cover that are facing each other.