A needleless syringe connector cap to be provided to a container having an aperture at its top, said cap comprising: an elastically deformable plastic valve body to be provided in or above said aperture at the top of the container, said valve body having a top surface and a bottom surface, wherein said valve body comprises a first passage and a second passage, each passage (37, 38) extending through the valve body from the top surface to the bottom surface thereof, and wherein said first passage comprises a first cross-slit valve in the extreme lower end of said passage, wherein said first cross-slit valve is embodied as an aeration valve, and wherein said second passage comprises a second cross-slit valve in the extreme lower end of said passage, wherein said second cross-slit valve is embodied as an aeration valve, and wherein the first passage is adapted to receive a first tapered syringe tip having a first diameter, e.g. a first Luer slip syringe tip or a first catheter syringe tip, in that said first tip is insertable into the first passage from the top surface of the valve body and in that the first passage seals around said first tip when said tip has been inserted into the first passage, and wherein the second passage is adapted to receive a second tapered syringe tip having a second diameter, e.g. a second Luer slip syringe tip or a second catheter syringe tip, in that said second tip is insertable into the second passage from the top surface of the valve body and in that the second passage seals around said second tip when said tip has been inserted into the second passage, and wherein when fluid is withdrawn from a container to which the cap is provided through the first passage using a first tapered syringe tip to which said first passage is adapted, the second cross-slit valve opens in response to an underpressure in said container resulting from said withdrawal, so as to aerate said container, and wherein when fluid is withdrawn from a container to which the cap is provided through the second passage using a second tapered syringe tip to which said second passage is adapted, the first cross-slit valve opens in response to an underpressure in said container resulting from said withdrawal, so as to aerate said container. an elastically deformable plastic valve body to be provided in or above said aperture at the top of the container, said valve body having a top surface and a bottom surface, wherein said valve body comprises a first passage and a second passage, each passage (37, 38) extending through the valve body from the top surface to the bottom surface thereof, and wherein said first passage comprises a first cross-slit valve in the extreme lower end of said passage, wherein said first cross-slit valve is embodied as an aeration valve, and wherein said second passage comprises a second cross-slit valve in the extreme lower end of said passage, wherein said second cross-slit valve is embodied as an aeration valve, and wherein the first passage is adapted to receive a first tapered syringe tip having a first diameter, e.g. a first Luer slip syringe tip or a first catheter syringe tip, in that said first tip is insertable into the first passage from the top surface of the valve body and in that the first passage seals around said first tip when said tip has been inserted into the first passage, and wherein the second passage is adapted to receive a second tapered syringe tip having a second diameter, e.g. a second Luer slip syringe tip or a second catheter syringe tip, in that said second tip is insertable into the second passage from the top surface of the valve body and in that the second passage seals around said second tip when said tip has been inserted into the second passage, and wherein when fluid is withdrawn from a container to which the cap is provided through the first passage using a first tapered syringe tip to which said first passage is adapted, the second cross-slit valve opens in response to an underpressure in said container resulting from said withdrawal, so as to aerate said container, and wherein when fluid is withdrawn from a container to which the cap is provided through the second passage using a second tapered syringe tip to which said second passage is adapted, the first cross-slit valve opens in response to an underpressure in said container resulting from said withdrawal, so as to aerate said container.