A pump having: a liquid pump comprising a piston-forming element reciprocally axially slidable in a piston chamber-forming body between a retracted position and an extended position defining a liquid compartment therebetween having a variable volume; an air pump comprising a flexible annular diaphragm member coaxially about the piston-forming element spanning between an axially outer piston end of the piston-forming element and the piston chamber-forming body to define a variable volume annular air compartment therebetween having a variable volume; a non-collapsible reservoir having an interior containing a fluid to be dispensed, the interior enclosed but for having an outlet port, the piston chamber-forming body closing the outlet port, a liquid inlet through the piston chamber-forming body from the interior of the reservoir to the liquid pump, whereby a retraction stroke of the piston forming element simultaneously forces air from the air compartment and liquid from the liquid compartment internally through an internal passageway of the piston-forming element and delivers the air and liquid from a dispensing outlet carried on the piston-forming element, and an extension stroke of the piston-forming element simultaneously draws the atmospheric air into the air compartment and the liquid from the interior of the reservoir into the liquid compartment via the liquid inlet, characterized by: an air relief passageway through the piston chamber-forming body providing communication between external atmospheric air and the interior of the reservoir,: the diaphragm member engaging the piston chamber-forming body to form therebetween an air relief valve across the air relief passageway to open and to close the air relief passageway dependent on the relative axial position of the piston-forming element and the piston chamber-forming body. a liquid pump comprising a piston-forming element reciprocally axially slidable in a piston chamber-forming body between a retracted position and an extended position defining a liquid compartment therebetween having a variable volume; an air pump comprising a flexible annular diaphragm member coaxially about the piston-forming element spanning between an axially outer piston end of the piston-forming element and the piston chamber-forming body to define a variable volume annular air compartment therebetween having a variable volume; a non-collapsible reservoir having an interior containing a fluid to be dispensed, the interior enclosed but for having an outlet port, the piston chamber-forming body closing the outlet port, a liquid inlet through the piston chamber-forming body from the interior of the reservoir to the liquid pump, whereby a retraction stroke of the piston forming element simultaneously forces air from the air compartment and liquid from the liquid compartment internally through an internal passageway of the piston-forming element and delivers the air and liquid from a dispensing outlet carried on the piston-forming element, and an extension stroke of the piston-forming element simultaneously draws the atmospheric air into the air compartment and the liquid from the interior of the reservoir into the liquid compartment via the liquid inlet, characterized by: an air relief passageway through the piston chamber-forming body providing communication between external atmospheric air and the interior of the reservoir,: the diaphragm member engaging the piston chamber-forming body to form therebetween an air relief valve across the air relief passageway to open and to close the air relief passageway dependent on the relative axial position of the piston-forming element and the piston chamber-forming body.