An adapter sleeve includes a main body having a through opening formed therethrough, so as to be assembled to a water pipe, the main body having an outer threaded portion formed on an outer surface thereof, the through opening having a first portion and a second portion, a diameter of the first portion of the through opening larger than that of the second portion, a step portion defined by the connecting site between the first portion and the second portion, the first portion having an inner threaded portion formed on an inner surface thereof, the second portion having a ring groove formed around an inner surface thereof, a ring member positioned on the ring groove.